คู่มือครู คณิตศาสตร์พื้นฐานอาชีพ ปวช

นาํ Warm-up สอน Instruction สสรรปุปุ WWrraapp--uupp
สรปุ
แบบฝกทักษะท่ี 3.2
17. ¼ÙŒÊ͹ÊÒ¸Ôµ¡ÒÃá¡ŒÊÁ¡ÒáíÒÅѧÊͧ
µÑÇá»Ãà´ÕÂǵÒÁ·Õè⨷¡íÒ˹´¨Ò¡ ค�ำช้ีแจง ใหผ้ ู้เรยี นแสดงวธิ ีทา� เพื่อหาค�าตอบของสมการตอ่ ไปน้ี โดยใชว้ ธิ กี ารแยกตัวประกอบ
Ẻ½ƒ¡·Ñ¡ÉзÕè 3.2 ãËŒ¼ÙŒàÃÕ¹´Ù໚¹ พร้อมท้งั ตรวจสอบค�าตอบ
µÑÇÍ‹ҧ
2
18. ¼ÙŒÊ͹ãËŒ¼ÙŒàÃÕ¹ᵋÅФ¹·íÒẺ½ƒ¡
·¡Ñ ÉзÕè 3.2 1. 2x + 11x - 21 = 0 ตรวจสอบค�ำตอบ........................ ว................ธิ........................ที ........................ำ� ................ ........................................ดห จ เ................น........ะงรั................อ่ืไ........นอื........ด........ง........ นั้........จ ้................ ........ำ........ ........ก........................ ................................ค(........2........2........ำx�................xต........2........ ................อ-+........................ บ ........3........1........ข........)1................อ(........x2........x........งxx........ ........ส........- +................ +-ม........ ........22 ........7ก........xxx713........)........ำ........ ................ร................ ........=======........ค........ ................อื........................ -00030........32................732 ........................ ................ห........................ร................จอื........................ะ ................ไ-................ด........7................ว้................ำ่................................................................................ แ แ ........2........ทท........(........2........นน-........(........7........คค23........)................ำ่่ำ2)2........ 2........ (........ +........xx49........+ ........29 ........ )1........== 1........1........ ++........1(........-23 ........(-........4733........ 23722........2 ใ33........))........ใน ........ ........น2-........----ส........ ........ส ........ม42222........ม........2........11011ก........ก........ ำ................ำร........======........ร........ ........ จ................จ000000........ะ........ ะ........ไ........ ........ไดส........ด................ว้ม........ว้........ำ่........ก........่ำ........ำ................ร................เ................ป........................น็ ................จ........................ร................งิ........................................

เกร็ดแนะ ผูสอน ........................................................................................................................ ...............9...8.. ..-.. .7...7.. ..-.. .2...1.. ...=.. ...0..................................
................................................................................................................................................................................................................................................ ด....งั......น....น้ั...... ....ค......่ำ....ข......อ....ง...... ..x...... ....ค....อื...... ....023.... .. ..ห..=.... ร......อื 0.... .... -..ส..7....ม......ก....ำ....ร....เ....ป......น็ ....จ......ร....งิ..........
¼ÊŒÙ ͹¤ÇáµÇÑ ÍÂÒ‹ §ÊÁ¡ÒáÒí Å§Ñ Êͧ
µÇÑ á»Ãà´ÂÕ Çº¹¡Ãдҹ 5 ÊÁ¡Òà áÅÇŒ ãËŒ ..........................................................................................................................................................................................................
¼àŒÙ ÃÂÕ ¹ÃÇ‹ Á¡¹Ñ ᡵÇÑ »ÃСͺ¢Í§ÊÁ¡ÒÃ
·§éÑ 5 ÊÁ¡Òù¹éÑ ËÒ¡ÊÁ¡ÒÃã´·è¼Õ ÙŒàÃÕ¹ 2
à¡´Ô ¤ÇÒÁʧÊÂÑ ãËŒ¼ŒÊ٠͹͸ºÔ ÒÂà¾ÔèÁàµÔÁ
¾ÃÍŒ Á¡µÇÑ ÍÂÒ‹ §ã˼Œ àŒÙ ÃÂÕ ¹à¡´Ô ¤ÇÒÁà¢ÒŒ 㨠2. x + 7x + 12 = 0 ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ�.. .....เ..น...อื่...ง..จ...ำ..ก... .........x...2.. ..+.. ..7...x.. ..+... ..1..2... .....=.. .....0... ..........จ...ะ..ไ..ด...ว้..ำ่............................................................................................

......... ............................................. จ .........ะ......ไ......ด......... ้.............................................(...x......... ......+......... ...3.........)......(......x...x... ...... +......+... ... ...4...x3...)...... ...............===...... ...............0-0.........3................................................................................................ แ......ท.........น(......-...ค...3......ำ่ )...... 2... ...x...+...9 ... ...= ...7...- ...(...- ...-...23...3...1 ...)...ใ ... ...น++...... ส... ...11...ม......202...ก... ......ำ...===...ร... ... ......จ000...... ะ...... ...ไส...ด......ม...ว้......กำ่......ำ......ร......เ......ป.........น็ ......จ.........ร......งิ...............
......... .............................................ดห .........งรั.........นอื......... นั้......... ...... ........................ค.........ำ�......ต............อ.........บ.........ข.........อ.........งx......ส... ......+ม...... ...ก...x4......ำ... ......ร...... ...==...ค... ......อื......... -0......-...4......3...... ......ห.........ร.........อื ......... ......-......4....................................... แ ......ท.........น(......-...ค...4......่ำ)...... 2... ...x1...+... ...6 ...=... 7... -...(...-... -...42...4... 8...)...ใ... ...น++.........ส ...11...ม......202...ก... ......ำ...===...ร... ... ......จ000...... ะ...... ...ไส...ด......ม...้ว......ก...่ำ...ำ......ร......เ......ป.........น็ ......จ.........ร......งิ...............

........................................................................................................................ด..งั...น..้นั... ..ค...ำ่ ..ข...อ..ง... .x... ..ค..ือ... ..-..3.. ..ห...ร..อื ... .-...4..........................

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84

Vขอ้ -สNอบEแTนว

¢ŒÍã´¤Í× ¼ÅÃÇÁ¢Í§¤íҵͺ¢Í§ÊÁ¡Òà x2 - 9x + 14 = 0
1. 9
2. 5
3. 1
4. -3
5. -5

84

นาํ Warm-up สอน Instruction สสรรปุปุ WWrraapp--uupp

2 สรปุ

3. x - x - 56 = 0 ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ�.. .....เ..น...อ่ื...ง..จ...ำ...ก.. ............x...2.. ..-.. ..x... ..-.. ..5..6... .....=.. .....0... ..........จ..ะ...ไ..ด...ว้..ำ่............................................................................................ 19. ¼ÊŒÙ ͹͸ºÔ Ò¶§Ö Ç¸Ô ¡Õ ÒõÃǨÊͺ¤Òí µÍº
¢Í§ÊÁ¡ÒáÒí ÅѧÊͧµÑÇá»Ãà´ÂÕ Ç
......... .............................................จ .........ะ.........ไ...ด............ ้ ..........................................(......x...... ......+......... ......7......)......(......x...x... ...... -......+... ... ...8...x7...)...... ...............===...... ...............00-.........7............................................................................................. แ.........ท.........น.........ค(......-...ำ่ ...7... ...)...x2......4 ......-=9...... ...(...-+...-...7 ...7... 7...)...ใ ... น...--...... ส ......55...ม...606......ก... ......ำ...===...ร... ... ......จ000...... ะ...... ไ...ส...ด......ม...ว้...ก...่ำ......ำ......ร......เ......ป......น็.........จ.........ร......งิ ...............
......... .............................................ด ห............งรั ......นอื......... นั้......... ...... ........................ค.........ำ�.........ต.........อ.........บ.........ข.........อ.........ง...x...ส... ......-ม...... ...ก...x8......ำ... ......ร...... ...==...ค... ......อื......... 80......-......7......... ......ห.........ร.........อื ......... ......8..........................................แ .........ท.........น.........ค.........่ำ...... ......x...... ...68...=...42... ... ...8...--... ... ...ใ88...น...... ...ส--...... ...ม...55......ก606.........ำ ......ร...===... ... ...จ......000ะ...... ไ...... ด...ส......ว้...ม...ำ่...ก.........ำ......ร......เ......ป......น็.........จ.........ร......งิ ............... 20. ¼ÙŒÊ͹ÊÒ¸Ôµ¡ÒõÃǨÊͺ¤íҵͺ¢Í§
ÊÁ¡ÒáÒí Å§Ñ ÊͧµÇÑ á»Ãà´ÂÕ Çã¹áºº½¡ƒ
.......................................................................................................................ด...ัง..น...ัน้... ..ค...ำ่ ..ข...อ..ง.. ..x... .ค...อื... ..-..7.. ..ห...ร..ือ.. ..8............................. ·¡Ñ ÉзÕè 3.2

.......................................................................................................................................................................................................... เกร็ดแนะ ผูสอน
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.......................................................................................................................................................................................................... ¼ÊŒÙ ͹¤ÇÃà´¹Ô ´¼Ù àŒÙ ÃÂÕ ¹ÃÐËÇÒ‹ §·Òí Ẻ
.......................................................................................................................................................................................................... ½ƒ¡·Ñ¡ÉÐ àÁè×ͼٌàÃÕ¹à¡Ô´¤ÇÒÁʧÊÑÂã¹
.......................................................................................................................................................................................................... Ẻ½ƒ¡·Ñ¡ÉТŒÍã´ ãËŒ¼ÙŒÊ͹䴌͸ԺÒÂ
.......................................................................................................................................................................................................... à¾èÔÁàµÁÔ ¨¹¼àŒÙ ÃÕ¹à¡Ô´¤ÇÒÁࢌÒã¨

2

4. y - 10y - 24 = 0 ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ�.. .....เ..น...อ่ื...ง..จ...ำ...ก.. ........y..2.. ..-.. ..1...0..y... ..-.. ..2..4... .....=.. .....0... ..........จ..ะ...ไ..ด...ว้..ำ่............................................................................................

...... ..............................จ ......ะ......ไ..ด........ ้ ........................(....y.... ....+...... ....2....)....(....y.... ....y-.... ....+1.... ..2..2..).... ..........==.... ..........00.................................................................... แ ......ท..(....น-....2..ค..)....ำ่2.. .. ..-..y.. .. 4..1=.. ..0.. +..(-.. ..-2..2..2.. 0..)ใ.. .. น..--....ส ....22..ม..44..ก.... ..ำ....==ร.... .. ..จ..00....ะ....ไ....ด......้ว....่ำ..............................................
y = -2....................................................................................................................... ......................................0... ...=.. ...0.. .. .ส...ม..ก...ำ..ร..เ..ป..น็...จ...ร..งิ .....
...... .............................. ห........ร....อื ...... ................................................................y...... ....-.... ....1....2..y.... ..........==.... ..........01......2..............................................................แ ......ท......น..1....ค2......ำ่21.. .. 4..-..y4.. .. .. 1=..-..0.. ..(1..1..1..22..2.. 0..)..ใ.. น..--....ส ....22..ม..44....ก.. ..ำ....==..ร.. .. ....จ00....ะ....ไ....ด......้ว....่ำ..............................................
... ...............ด....งั ..น...นั้... .. ........ค...ำ�...ต...อ...บ...ข...อ...ง..ส...ม...ก...ำ...ร.. ..ค...อื... ..-..2... ..ห...ร...อื ... ..1..2............ ......................................0... ...=.. ...0.. .. .ส...ม..ก...ำ..ร..เ..ป..น็...จ...ร..งิ .....

.......................................................................................................................ด...ัง..น...นั้... ..ค...ำ่ ..ข...อ..ง.. ..y.. ..ค...ือ.. ..-..2... .ห...ร..ือ... ..1..2...........................

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85

ÇÔà¤ÃÒÐˏ¤íҵͺ
Ç¸Ô Õ·íÒ à¹Í×è §¨Ò¡ x2 - 9x + 14 = 0 ¨Ðä´ÇŒ ‹Ò

(x - 7)(x - 2) = 0
¨Ðä´Œ x - 7 = 0

x=7
ËÃÍ× x - 2 = 0

x=2
´Ñ§¹Ñ¹é ¤íҵͺ¢Í§ÊÁ¡Òà ¤Í× 2 áÅÐ 7 áÅÐ

¼ÅÃÇÁ¢Í§¤íҵͺÊÁ¡Òà ¤×Í 2 + 7 = 9

µÍº¢ŒÍ 1.

85

นาํ Warm-up สอน Instruction สสรรปุปุ WWrraapp--uupp
สรปุ
2
21. ¼ÙŒÊ͹ÊØ‹ÁµÑÇá·¹¼ÙŒàÃÕ¹ÍÍ¡ÁÒáÊ´§
Ç¸Ô ·Õ Òí ¡ÒÃá¡ÊŒ Á¡ÒáÒí Å§Ñ ÊͧµÇÑ á»Ãà´ÂÕ Ç 5. 30x + 7x - 4 = 0 ตรวจสอบค�ำตอบ................................. ว ......................ธิ......................ท...........ี .................................ำ� ...................... ....................................................... จ เ หด......................น...........ะงรั......................อ่ืไ...........นอื ...........ด...........ง........... นั้ ...........จ ้...................... ...........ำ........... ...........ก......................(........... ...........1......................5...........ค......................x...........3ำ�........... ...........ต0...........-......................x...........อ ......................24บ........... ...........)...........+...........ข(...........1........... 2อ......................275x...........ง...........xxx........... ...........ส........... +...........1...........+--ม ...........52........... 1...........ก...........xxxx441...........)...........ำ........... ...........ร................................. ========......................ค........... ......................อื................................. 00400-1-...........42...........11...........1 54......................5...................... ......................ห.................................รจ.................................อืะ................................. ไ...........ด...........-...........2...........1...........ว้ ......................ำ่................................................................................................... แ แ ด .................................ทท33ัง......................00น...........นน......................((ั้น...........คค1...........-...........24 ...........่ำำ่15...........ค........... ))...........13...........xx22่ำ........... 52........... ข13...........++24...........== 05...........อ ...........+ ...........77 ง........... 1-...........--24(( ...........261115........... 1-x...........245508-- ...........1522........... ใใ 77...........ค))...........นน--0........... ........... อื --...........--สส...........1661 ........... ...........=มม50508228404...........1........... 4กก........... 5......................0ำำ ...........=======........... หรร........... ........... ...........รส...........0000000จจ...........อื...........ม ...........ะะ........... ...........กสไไ...........-...........ดด2ำ...........ม1...........ร...........ว้ว้ ก......................เ่ำ่ำ...........ำป......................ร...........น็ ...........เ......................จป......................รน็......................งิ...........จ.................................ร......................งิ .......................................................
ãºáºº½ƒ¡·¡Ñ Éк¹¡Ãдҹ
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22. ¼ÙŒÊ͹áÅмٌàÃÕ¹ËÇÁ¡Ñ¹µÃǨÊͺ ..........................................................................................................................................................................................................
¤ÇÒÁ¶Ù¡µŒÍ§¢Í§¡ÒÃáÊ´§ÇÔ¸Õ·íÒ¡Òà ..........................................................................................................................................................................................................
á¡ŒÊÁ¡ÒáÒí ÅѧÊͧµÇÑ á»Ãà´ÕÂÇ
2
เกร็ดแนะ ผูสอน
6. 9x - 16 = 0 ตรวจสอบค�ำตอบ...ว..ธิ..ท.ี ...ำ� .. .....เ..น...อ่ื ...ง..จ...ำ..ก... ..................9...x...2. ..-... ..1..6... ....=... .....0... .........จ...ะ...ไ.ด....ว้ ..ำ่............................................................................................
¼ÙŒÊ͹¤ÇÃÊÃØ»ÇÔ¸Õ¡ÒèѴÃÙ»µ‹Ò§æ
¢Í§ÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ãà´ÕÂÇãËŒ .................. .......................................................................................... ห จ ..................ะร............ไ......อื ......ด............ ...... ้............................................................(............3..................x............ ............+.................. ............4(............)3............(......x......3......33)......x......x2x............ ...... --............+- ......3...... 44............xx44......)2............ ........................======.................. ..............................00-00-......3......4......4................................................................................................................................................................................................ แ แ ............ทท..................นน..................คค..................ำ่ำ่ ............ ............xx............ ......9......==......99(...... ......((-......-433......13434......941......6 ......))......6 ใ)22......ใน...... ......น......----ส............ ส......ม1111............ม......66066ก............ก ......ำ......ำ......ร=====......ร...... ............ จ......00000จ......ะ...... ......ะ......ไ ......สได............ด......ม้ว............้ว่ำ......ก......่ำ......ำ............ร............เ............ป............น......็ ............จ..................ร............งิ ..............................
¼ÙŒàÃÕ¹´Ù à¾è×ͼٌàÃÕ¹¨Ð䴌ᡌÊÁ¡ÒÃËÒ ......... .............................................ด .........งั.........น.........น้ั ......... ...... ........................ค.........ำ�......ต............อ......บ............ข......อ.........ง.........ส.........ม...3......ก...xx......ำ... ...ร......... ==......ค... ......อื......... 443......±............34....................................................................................... ด ......ัง.........น.........นั้ ...... ......ค.........่ำ......ข.........9อ......(ง......1... 91...6x......6)... ... ...ค...--...อื... ...11... ......660±...... ...34......===...... .........000...... ...... ...ส.........ม.........ก......ำ......ร......เ......ป......น...็ ......จ.........ร......งิ ...............
¤Òí µÍºä´§Œ Ò‹ ¢¹Öé
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86

ขVอ้ -สNอบEแTนว

x + 4 ໚¹µÇÑ »ÃСͺµÑÇË¹Ö§è ¢Í§¾Ë¹Ø ÒÁ㹢͌ ã´
1. x2 - 4x + 4
2. x2 + 2x - 3
3. x2 + 8x + 16
4. x2 + 10x + 25
5. x2 - 6x + 8

86

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2 สรปุ

7. 1 - 81x = 0 ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ� .. .....เ..น...อื่ ...ง..จ...ำ..ก... ..................1... ..-.. ..8...1..x...2.. ....=... .....0... .........จ...ะ...ไ..ด...ว้ ..ำ่............................................................................................ 23. ¼ÙŒÊ͹áÅмٌàÃÕ¹µÃǨÊͺ¤íҵͺáÅŒÇ
ËÒ¡¾ºÇ‹ÒµÑÇá·¹¼ÙŒàÃÕ¹áÊ´§ÇÔ¸Õ·íÒ
.................. .......................................................................................... จ ห..................ะร............ไ......อื ......ด............ ...... ้............................................................(......9..................x............ ............+.................. ............1............)......8......(............91............99......xx......xx...... 2............ -............+-- ............9 1............xx111......)............ ........................======.................. ..............................000-0-......9......1......1................................................................................................................................................................................................ แแ ............ทท..................นน..................คค..................ำ่่ำ............ ............xx............ ......1......==1......1 ...... ...... - ......-91-......- 9......1...... 8 ......8......ใ 81......น......ใ111......(น......(ส -(......9......8-ส911......1ม...... 1......ม))......10ก)22............ก ำ............ำ......ร=====............ร ............จ ......00000......จะ...... ......ะไ...... ......ดสไ............ด......ว้ม............ว้ำ่......ก......ำ่......ำ............ร............เ............ป..................น็ ............จ..................ร............งิ.............................. äÁ¶‹ ¡Ù µÍŒ § ¼ÊŒÙ ͹áÅмàŒÙ ÃÂÕ ¹ªÇ‹ ¡¹Ñ á¡äŒ ¢
......... ............................................. ด.........งั.........น.........น้ั ......... ...... ........................ค.........ำ�......ต............อ......บ............ข.........อ......ง.........ส.........ม......9...ก...xx......ำ... ...ร......... ==......ค... ......อื......... -91......±1............91....................................................................................... ด .........งั ......น.........้นั ...... ......ค.........ำ่......ข......1...อ...... ง...-...... ... x...8...... ...11...ค0...( ...ือ... 8-...1... 1 ...=...1±)... ... ...91...0...==... ... ... ......ส00......ม.........ก.........ำ......ร......เ...ป.........น็.........จ......ร......งิ................................. ã˶Œ ¡Ù µÍŒ §

.......................................................................................................................................................................................................... เกร็ดแนะ ผูสอน
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.......................................................................................................................................................................................................... ¼ÙŒÊ͹¤ÇêÕéá¹ÐàÃ×èͧà¤Ã×èͧËÁÒ·Õè
.......................................................................................................................................................................................................... ÍÂã‹Ù ¹ÊÁ¡Òà ¼ŒàÙ ÃÂÕ ¹¤ÇþԨÒóÒáÅÐá¡Œ
ÊÁ¡ÒÃÍÂÒ‹ §Ãͺ¤Íº ËÒ¡à¤ÃÍ×è §ËÁÒÂã¹
2 ÊÁ¡Òü´Ô ¨Ð·Òí ãˌ䴌¤Òí µÍº¢Í§ÊÁ¡Ò÷èÕ
¤ÅÒ´à¤ÅÍ×è ¹ä»¨Ò¡à´ÁÔ áÅÐàÁÍ×è á·¹¤Ò‹ ã¹
8. x + 14x + 49 = 0 ตรวจสอบค�ำตอบ...ว..ธิ..ท.ี ...ำ� .. .....เ..น...อื่ ...ง..จ...ำ..ก... .......x..2.. ..+.. ..1...4..x... ..+... ..4..9... ....=... .....0... .........จ...ะ...ไ.ด....ว้ ..ำ่............................................................................................ ÊÁ¡ÒÃáÅÇŒ ¨Ð·Òí ãËÊŒ Á¡ÒÃäÁ‹à»¹š ¨Ã§Ô

......... ............................................. จ .........ะ......ไ......ด......... ้..........................................(......x......... ......+...... ......7.........)......(...x......x... ...... +......+ ...... 7......x7...)...... ............===......... ...............00-......7................................................................................................ แ.........ท...(......-น......7...ค...)......2ำ่ ... ... ...+4...x... ...9 ...1...= ...4...- ...(...- ...-...97...7...8 ...)...ใ ... น...++...... ส ......44...ม...909......ก... ......ำ...===...ร... ... ......จ000...... ะ...... ไ...ส...ด......ม...ว้...ก...ำ่......ำ......ร......เ......ป......น็.........จ.........ร......งิ ...............
... ...............ด...งั...น...น้ั ... .. ........ค...ำ� ..ต....อ..บ....ข..อ...ง...ส...ม...ก...ำ..ร... ..ค...อื... ..-..7..............................ด...ัง..น...ั้น... ..ค...ำ่ ..ข...อ..ง.. ..x... .ค...อื... ..-..7..........................................

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87

ÇàÔ ¤ÃÒÐˏ¤Òí µÍº
ÇÔ¸·Õ Òí ᡵÇÑ »ÃСͺ¢Í§¾ËعÒÁ·Ø¡µÑÇàÅÍ× ¡ ¨ÐàËç¹ä´ÇŒ ‹Ò

µÇÑ àÅÍ× ¡·Õè 1 x2 - 4x + 4 = (x - 2)(x - 2)
µÇÑ àÅÍ× ¡·Õè 2 x2 + 2x - 3 = (x + 3)(x - 1)
µÇÑ àÅÍ× ¡·èÕ 3 x2 + 8x + 16 = (x + 4)(x + 4)
µÑÇàÅ×Í¡·èÕ 4 x2 - 10x + 25 = (x - 5)(x - 5)
µÇÑ àÅÍ× ¡·Õè 5 x2 - 6x + 8 = (x - 4)(x - 2)
´§Ñ ¹Ñ¹é x + 4 ໹š µÇÑ »ÃСͺµÑÇ˹§èÖ ¢Í§ x2 + 8x + 16

µÍº¢ŒÍ 3.

87

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สรปุ
2
24. ¼ÙŒÊ͹áÅмٌàÃÕ¹ËÇÁ¡Ñ¹ÊÃØ»¡ÒÃá¡Œ
ÊÁ¡ÒáÒí Å§Ñ ÊͧµÇÑ á»Ãà´ÂÕ Ç â´ÂÇ¸Ô ¡Õ Òà 9. x + 6x + 5 = 0 ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ�.. .....เ..น...อ่ื...ง..จ...ำ...ก.. ............x...2. ..+... ..6..x... ..+.. ..5... ....=... .....0... ..........จ..ะ...ไ..ด...ว้ ..ำ่............................................................................................
ᡵÑÇ»ÃСͺ
......... ............................................. จ.........ะ.........ไ...ด......... ...้ ..........................................(...x......... ......+......... ...5.........)......(......x...x... ...... +......+... ... ...1...x5...)...... ............===......... ...............0-0......5................................................................................................แ .........ท.........น.........(ค...-......ำ่ 5...... )......x2... ...2 ...+...=5... ... ...6...--...(...5 ...-...3 ...5...ใ0...)น... ... ...ส++...... ... ม...550......ก... ...ำ......===...ร... ... ......จ000...... ะ... ...ไ...ส...ด......ม...ว้ ...ก...ำ่......ำ......ร......เ...ป.........น็.........จ......ร.........งิ ...............
25. ¼ÙŒÊ͹»ÃÐàÁÔ¹¼Å¡ÒÃàÃÕ¹ÃÙŒ¨Ò¡¡Òà ......... ............................................. หด............งรั ......นอื......... น้ั......... ...... ........................ค.........ำ�......ต............อ.........บ.........ข.........อ.........งx......ส... ......+ม...... ...ก...x1......ำ... ......ร...... ==......ค... ......อื......... 0-......-1......5......... ......ห.........ร.........อื ...... ......-.........1.................................... แ.........ท.........น.........(ค...-......ำ่ 1...... )......x2... ... ...+...=... ... ...16...-... (...1...--... ... 1...ใ6...)น...... ...ส++......... ม...550......ก... ......ำ...===...ร... ... ......จ000...... ะ... ...ไ...ส...ด......ม...้ว...ก...ำ่......ำ......ร......เ...ป.........น็.........จ......ร.........งิ ...............
µÃǨÊͺ¡Ò÷Òí ¡¨Ô ¡ÃÃÁáÅСÒõÃǨ
Ẻ½ƒ¡·¡Ñ ÉТͧ¼àŒÙ ÃÕ¹ .......................................................................................................................ด...งั..น...้ัน... ..ค...่ำ..ข...อ..ง.. ..x.. ..ค...ือ... ..-..5.. ..ห...ร..อื .. ..-..1...........................

เกร็ดแนะ ผสู อน ..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
㹡ÒõÃǨẺ½¡ƒ ·¡Ñ ÉТͧ¼àŒÙ ÃÂÕ ¹ ..........................................................................................................................................................................................................
¼ÙŒÊ͹¤ÇÃÊÃØ»¢ŒÍ¼Ô´¾ÅҴ㹡ÒÃá¡Œ ..........................................................................................................................................................................................................
ÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ãà´ÕÂǢͧ¼ÙŒàÃÕ¹ ..........................................................................................................................................................................................................
·ÕèÁÕÁÒ¡·ÕèÊØ´ áÅŒÇãËŒ¼ÙŒÊ͹¹íÒä»Í¸ÔºÒ ..........................................................................................................................................................................................................
㹪¹éÑ àÃÂÕ ¹ à¾Í×è ·¼Õè àŒÙ ÃÂÕ ¹¨Ðä´ÁŒ ¤Õ ÇÒÁà¢ÒŒ ã¨
ÍÂÒ‹ §¡ÃШҋ § 2

10. 3x + 2x - 8 = 0 ตรวจสอบค�ำตอบ...ว...ธิ ..ที...ำ�... .....เ.น...อ่ื...ง...จ..ำ...ก... .........3..x...2.. ..+.. ..2...x.. ..-... ..8.. .....=.. .....0... ..........จ...ะ..ไ..ด...ว้..ำ่............................................................................................

........................... ................................................................................................................................................ ดห จ ..................ะ.........งรั ..................ไนอื..................ด......... น้ั.................. ้ .................. .................. ...............................................................ค.........(.........3.........ำ�..................x.........ต......... ..................อ-........................... บ..................4.........ข.........)..................อ(.........3.........x.........ง.........xx .........ส......... +..................+-ม......... .........3......... 2.........ก.........xxx24).........ำ......... ..................ร.................. .........======.........ค......... ..................อ.........ื .................. -0400.........34..................243........................... ..................ห...........................ร..................อื........................... ..................-...........................2.............................................................................................................................. ด แแ ..................ททงั...........................นนน.........3.........น้ั.........คค(.........3-......... .........ำ่่ำ(.........2ค.........43 3.........).........xxำ่2)(.........1......... 2 ข31......... 69+.........==.........6อ1+......... .........)2 ง2+.........-34 .........2 .........( 2+......... -(-x.........83 ใ.........43 2......... ใน 83.........4ค)).........น-.........ส ......... ือส.........----.........ม2......... 3ม.........4800888ก.........43.........ก......... ำ ..................ำหร.........=======.........ร ......... ร......... จ.........ือ.........จ0000000.........ะ......... ะ.........ไ......... -.........ไดสส.........2ด..................ว้มม.........ว้.........่ำกก.........ำ่..................ำำ..................รร..................เเ..................ปป..................นน็็...........................จจ..................ร.........ร..................งงิิ .............................................

..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................

88

Vข้อ-สNอบEแTนว

¤Òí µÍº¢Í§ÊÁ¡Òà 16x2 - 625 = 0 µÃ§¡Ñº¢ŒÍã´
1. 0
2. 245
3. - 245
4. ± 245
5. 16

88

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นาํ

2) การแก้สมการก�าลังสองตวั แปรเดียวโดยใชส้ ตู ร 1. ¼ÊŒÙ ͹à¢ÂÕ ¹ÊÁ¡ÒáÒí Å§Ñ ÊͧµÇÑ á»Ãà´ÂÕ Ç·èÕ
ในกำรแก้สมกำรก�ำลังสองตัวแปรเดียวโดยใช้สูตร ท�ำได้โดยพิจำรณำจำกรูปท่ัวไปของ äÁÊ‹ ÒÁÒöᡵÇÑ »ÃСͺ䴺Œ ¹¡Ãдҹ
áÅÇŒ «Ñ¡¶ÒÁ¼ÙàŒ ÃÂÕ ¹ ´§Ñ ¹Õé
สมกำรกำ� ลงั สองตัวแปรเดยี ว ax2 + bx + c = 0 เมื่อ a และ b เปนสัมประสิทธิ์ของตวั แปรใน • ¼ÙŒàÃÕ¹ÁÕá¹Ç¤Ô´ËÃ×ÍÇÔ¸Õã´ã¹¡ÒÃËÒ
สมกำรซ่ึง a ≠ 0 และ c เปน คำ่ คงตวั เรำสำมำรถจัดรูปใหมเ่ พือ่ หำค่ำของตวั แปร x ได้ ดังน้ี ¤íҵͺ

จำก ax2 + bx + c = 0 จะได้ว่ำ á¹ÇµÍº
ax2 + bx = -c ÊÒÁÒöËÒ¤íҵͺ䴌â´Â¡ÒÃá¡ŒÊÁ¡ÒÃ
xa2 + bax = -ac ¡íÒÅѧÊͧµÇÑ á»Ãà´ÂÕ Çâ´ÂãªÊŒ ÙµÃ
x2 + bax = -ac
สอน
x2 + bax + (2ba)2 = -ac + (2ba)2
2. ¼ÊŒÙ ͹ã˼Œ àŒÙ ÃÂÕ ¹È¡Ö ÉÒ¡ÒÃá¡ÊŒ Á¡ÒáÒí ŧÑ
[x + 2ba]2 = - ac + 4ba22 ÊͧµÇÑ á»Ãà´ÕÂÇâ´Â㪌Êٵà ã¹Ë¹Ñ§Ê×Í
[x + 2ba]2 = - -4a4ca+2 b2
[x + 2ba]2 = - b2 4-a42ac 3. ¼ÙŒÊ͹͸ԺÒÂà¾èÔÁàµÔÁà¡ÕèÂǡѺ¡ÒÃá¡Œ
ÊÁ¡ÒáÒí Å§Ñ ÊͧµÑÇá»Ãà´ÂÕ Çâ´Â㪌ÊÙµÃ
x + 2ba = ± b2 4-a42ac
x + 2ba = ± b2 -2a4ac 4. ¼ÙŒÊ͹͸ԺÒÂà¡ÕèÂǡѺÊٵ÷èÕ㪌㹡ÒÃ
á¡ŒÊÁ¡ÒÃÊͧµÑÇá»Ãà¾è×ÍËÒ¤íҵͺã¹
x = - 2ba ± b2 -2a4ac ¡Ãͺ Math Notes
ดงั นนั้ x = -b ± 2ba2 - 4ac

Math Notes

จำกสตู รทใ่ี ชใ้ นกำรแกส้ มกำรกำ� ลงั สองตวั แปรเดยี วเพอื่ หำคำ� ตอบ หำกพจิ ำรณำคำ� ตอบของสมกำร เกร็ดแนะ ผูส อน
พบวำ่
¼ÙŒÊ͹¤ÇáµÑÇÍ‹ҧÊÁ¡ÒáíÒÅѧ
ถำ้ b2 - 4ac > 0 แลว้ ค�ำตอบของสมกำรจะเปนจำ� นวนจรงิ 2 ค่ำ ÊͧµÑÇá»Ãà´ÕÂÇ áÅŒÇà¢Õ¹º¹¡Ãдҹ
ถ้ำ b2 - 4ac = 0 แล้วค�ำตอบของสมกำรจะเปน จ�ำนวนจรงิ 1 คำ่ ¨Ò¡¹Ñé¹ãËŒ¼ÙŒàÃÕ¹¾Ô¨ÒóÒÇ‹ÒÁÕÊÁ¡ÒÃ
ถ้ำ b2 - 4ac < 0 แลว้ สมกำรจะไมม่ ีคำ� ตอบท่เี ปน จ�ำนวนจรงิ ã´ºŒÒ§·ÕèÊÒÁÒöᡵÑÇ»ÃСͺ䴌
áÅÐÁÕÊÁ¡ÒÃã´ºŒÒ§·ÕèäÁ‹ÊÒÁÒöá¡
89 µÑÇ»ÃСͺ䴌 à¾è×Í໚¹à·¤¹Ô¤ãˌᡋ
¼àŒÙ ÃÂÕ ¹ã¹¡Òþ¨Ô ÒóÒá¡ÊŒ Á¡ÒáÒí Å§Ñ Êͧ
µÑÇá»Ãà´ÕÂÇ

ÇÔà¤ÃÒÐˏ¤íҵͺ

ÇÔ¸·Õ Òí ¨Ò¡ÊÁ¡ÒÃ 16x2 - 625 = 0 ผูเรียน ควรรู
¨Ðä´Œ (4x)2 - (25)2 = 0
ÊÁ¡ÒèÐäÁÁ‹ ¤Õ Òí µÍº·èÕ໚¹¨íҹǹ¨ÃÔ§ ËÁÒ¶§Ö ÊÁ¡ÒáíÒÅѧÊͧ
(4x + 25)(4x - 25) = 0 µÑÇá»Ãà´ÕÂÇ·ÕèàÁ×èÍá¡ŒÊÁ¡ÒÃâ´Â㪌ÊÙµÃáÅŒÇ ¨íҹǹã¹ÃÒ¡·ÕèÊͧµÔ´Åº
·Òí ãËäŒ ÁÊ‹ ÒÁÒö¶Í´ÃÒ¡·ÊÕè ͧÍÍ¡ÁÒä´Œ ¨§Ö ·Òí ãËÊŒ Á¡Òù¹éÑ äÁÁ‹ ¤Õ Òí µÍº
¨Ðä´Œ 4x + 25 = 0 ᵋËÒ¡ã¹ÃдѺªé¹Ñ àÃÂÕ ¹·èÕʧ٠¢Ö¹é ¨ÐÊÒÁÒöá»Å§¨íҹǹã¹ÃÒ¡·ÕèµÔ´Åº
x = -245 ໚¹¨íҹǹ¨Ô¹µÀҾ䴌

ËÃÍ× 4x - 25 = 0
x = 245
´§Ñ ¹Ñé¹ ¤Òí µÍº¢Í§ÊÁ¡Òà ¤Í× ± 245
µÍº¢ÍŒ 4.

89

นาํ Warm-up สสออนน InIsntsrtruuccttioionn สรปุ Wrap-up
สอน
ตัวอยำงท่ี 16 ตรวจสอบค�ำตอบ
5. ¼ÙŒÊ͹ãËŒ¼àŒÙ ÃÂÕ ¹È¡Ö ÉÒµÇÑ Í‹ҧ·èÕ 16 áÅÐ
µÑÇÍÂÒ‹ §·Õè 17 ¨Ò¡Ë¹Ñ§Ê×ÍàÃÕ¹ จงหาคา� ตอบของสมการ x2 - 12x + 11 = 0 แทนคำ่ x = 11 ในสมกำร
วธิ ีทา� จำกสมกำร x2 - 12x + 11 = 0 จะไดว้ ่ำ 112 - 12(11) + 11 = 0
6. ¼ÙŒÊ͹͸ԺÒÂà¾ÔèÁàµÔÁà¡ÕèÂǡѺµÑÇÍ‹ҧ·èÕ 121 - 132 + 11 = 0
16 áÅеÑÇÍ‹ҧ·èÕ 17 ¨Ò¡Ë¹§Ñ Ê×ÍàÃÕ¹ a = 1, b = -12 และ c = 11 0 =0
-b ± b2 - 4ac
7. ¼ÊŒÙ ͹¡µÇÑ ÍÂÒ‹ §â¨·Â» Þ˜ ËÒà¡ÂÕè Ç¡ºÑ ¡Òà จำก x = 2a จะได้ว่ำ แทนค่ำ x = 1 ในสมกำร
á¡ŒÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ãà´ÕÂÇâ´Â㪌 12 - 12(1) + 11 = 0
ʵ٠à 5 ¢ŒÍ â´ÂµÍŒ §ÁÕû٠Ẻ¢Í§ÊÁ¡Òà x = -(-12) ± (-12)2 - 4(1)(11) 1 - 12 + 11 = 0
໚¹ÃٻẺà´ÕÂǡѺÊÁ¡ÒÃã¹µÑÇÍ‹ҧ·èÕ 2(1) 0 =0
16 áÅеÑÇÍ‹ҧ·èÕ 17 à¢ÂÕ ¹º¹¡Ãдҹ 12 ± 144 - 44
x = 2 ดงั นั้น ค่ำของ x คอิื 11 หรือ 1
8. ¼ÊŒÙ ͹ÊÁ‹Ø ¼àŒÙ ÃÂÕ ¹ÍÍ¡ÁÒáÊ´§Ç¸Ô ·Õ Òí áÅÐËÒ
¤Òí µÍºË¹ÒŒ ªÑé¹àÃÕ¹ x = 12 ± 100 ตอบ
x = 12 ±210
9. ¼ÊŒÙ ͹áÅмàŒÙ ÃÂÕ ¹¤¹Í¹è× æ ÃÇ‹ Á¡¹Ñ ¾¨Ô ÒÃ³Ò x = +2
áÅеÃǨÊͺ¤ÇÒÁ¶Ù¡µŒÍ§¢Í§¢éѹµÍ¹ 12 2 10, 12 - 10
¡ÒÃáÊ´§ÇÔ¸·Õ Òí áÅФíҵͺ·Õäè ´Œ 2
x = 11, 1
เกร็ดแนะ ผสู อน ดงั นนั้ ค�ำตอบของสมกำร คอื 11 หรือ 1

¼ÙŒÊ͹à¢Õ¹ÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ã ตัวอยำงท่ี 17 ตรวจสอบค�ำตอบ
à´ÂÕ Çº¹¡Ãдҹ áÅŒÇã˼Œ àŒÙ ÃÂÕ ¹¾¨Ô ÒóÒ
ÇÒ‹ ¤Ò‹ a, b áÅÐ c Á¤Õ ‹Òà·‹Ò¡ºÑ à·‹Òã´ จงหาคา� ตอบของสมการ 2x2 - 2x - 40 = 0 แทนคำ่ x = 5 ในสมกำร
à¾×èÍãËŒ¼ÙŒàÃÕ¹ÊÒÁÒö᷹¤‹Òã¹ÊÙµÃä´Œ วิธีทา� จำกสมกำร 2x2 - 2x - 40 = 0 จะไดว้ ่ำ 2(5)2 - 2(5) - 40 = 0
ÍÂÒ‹ §¶¡Ù µŒÍ§ 50 - 10 - 40 = 0
a = 2, b = -2 และ c = 40 0 =0
-b ± b2 - 4ac
จำก x = 2a จะได้ว่ำ แทนคำ่ x = -4 ในสมกำร
2(-4)2 - 2(-4) - 40 = 0
x = -(-2) ± (-2)2 - 4(2)(40) 32 + 8 - 40 = 0
2(2) 0 =0
2 ± 4 - 320
x = 4 ดงั นนั้ คำ่ ของ x คอืิ 5 หรือ -4

x = 2 ± 324 ตอบ
= 2 ±418
x = +44-418,
= 2 2 -418
x
x 5,

ดังนนั้ คำ� ตอบของสมกำร คือ 5 หรอื -4

90

ขV้อ-สNอบEแTนว

¢ŒÍã´à»¹š ¤Òí µÍº¢Í§ÊÁ¡Òà 3x2 + 7x - 6 = 0
1. -1 áÅÐ 1
13412123
2. -3 áÅÐ
3. -5 áÅÐ
4. -7 áÅÐ
5. -9 áÅÐ

90

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ตัวอยำงท่ี 18 สอน

จงหาคา� ตอบของสมการ 3x2 - 2x + 2 = 0 10. ¼ÊŒÙ ͹ã˼Œ àŒÙ ÃÂÕ ¹È¡Ö ÉÒµÇÑ ÍÂÒ‹ §·Õè 18 áÅÐ
วธิ ีทา� จำกสมกำร 3x2 - 2x + 2 = 0 จะได้วำ่ µÑÇÍÂÒ‹ §·èÕ 19 ¨Ò¡Ë¹§Ñ Ê×ÍàÃÂÕ ¹

a = 3, b = -2 และ c = 2 b2 - 4ac 11. ¼ÊŒÙ ͹͸ºÔ ÒÂà¾ÁÔè àµÁÔ à¡ÂèÕ Ç¡ºÑ µÇÑ ÍÂÒ‹ §·èÕ
-b ± 2a 18 áÅеÇÑ Í‹ҧ·èÕ 19 ¨Ò¡Ë¹Ñ§Ê×ÍàÃÂÕ ¹
จำก x = จะได้วำ่
12. ¼ÙŒÊ͹¡µÑÇÍ‹ҧ⨷»˜ÞËÒà¡ÕèÂǡѺ
x = -(-2) ± (-2)2 - 4(3)(2) ¡ÒÃá¡ŒÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ãà´ÕÂÇ
2(3) â´ÂãªÊŒ ٵà 5 ¢ÍŒ â´ÂµÍŒ §ÁÃÕ Ù»áºº¢Í§
4 ± 4 - 24 ÊÁ¡ÒÃ໚¹ÃٻẺà´ÕÂǡѺÊÁ¡ÒÃã¹
x = 6 µÇÑ ÍÂÒ‹ §·èÕ 18 áÅеÑÇÍÂÒ‹ §·èÕ 19 à¢Õ¹
º¹¡Ãдҹ
x = 4 ± -20
4 13. ¼ÙŒÊ͹ÊØ‹Á¼ÙŒàÃÕ¹ÍÍ¡ÁÒáÊ´§ÇÔ¸Õ·íÒáÅÐ
เน่ืองจำก b2 - 4ac < 0 ËÒ¤Òí µÍºË¹ÒŒ ªÑ¹é àÃÕ¹

ดังน้นั สมกำรนไี้ ม่มีคำ� ตอบ ตอบ 14. ¼ÙŒÊ͹áÅмٌàÃÕ¹¤¹Íè×¹æ ËÇÁ¡Ñ¹
¾Ô¨ÒóÒáÅеÃǨÊͺ¤ÇÒÁ¶Ù¡µŒÍ§
ตัวอยำงที่ 19 ตรวจสอบค�ำตอบ ¢Í§¢éѹµÍ¹¡ÒÃáÊ´§ÇÔ¸Õ·íÒáÅФíҵͺ
·èäÕ ´Œ
จงหาค�าตอบของสมการ 10x2 = 17x - 3 แทนค่ำ x = 23 ในสมกำร
10x2 = 17x - 3 10( 32 )2 เกร็ดแนะ ผูส อน
วธิ ีทา� จำกสมกำร 17(23) - 3
จะได้ 10x2 - 17x + 3 = 0 จะได้ว่ำ = ¼ÙŒÊ͹ÍҨ¡µÑÇÍ‹ҧ⨷à¡ÕèÂǡѺ
ÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ãà´ÕÂÇà¾èÔÁàµÔÁº¹
a = 10, b = -17 และ c = 3 22.5 = 22.5 ¡Ãдҹ áÅÇŒ ã˼Œ àŒÙ ÃÂÕ ¹ÃÇ‹ Á¡¹Ñ áÊ´§Ç¸Ô ·Õ Òí
0 =0 â´ÂãªÇŒ ¸Ô ¡Õ ÒöÒÁ - µÍº ¡Ñº¼ŒàÙ ÃÂÕ ¹ à¾×èÍ
จำก x = -b ± b2 - 4ac จะได้ว่ำ 51 µÃǨÊͺ¤ÇÒÁà¢ÒŒ 㨢ͧ¼àŒÙ ÃÂÕ ¹ 㹡ó·Õ èÕ
2a แทนคำ่ x = ในสมกำร ¼ÙŒàÃÂÕ ¹à¡´Ô ¢ÍŒ ʧÊÑ ã˼Œ ŒÊ٠͹¡µÑÇÍÂÒ‹ §
-(-17) ± (-17)2 - 4(10)(3) 10( 15 )2 à¾ÔÁè àµÁÔ ÍÕ¡ 1 ¢ŒÍ
x = 2(10) = 17(51) - 3

x = 17 ± 289 - 120 0.4 = 0.4
20 0 =0
ดงั นน้ั คำ�xxxxตอบ====ของ113223ส7700,ม2,+±ก20512ำ014ร0136,ค9ือ17322-0=ห13รอื 17152±0 13 ดังนน้ั คำ่ ของ x คืิอ 23 หรือ 15

ตอบ

91

ÇàÔ ¤ÃÒÐˏ¤íҵͺ ผูเรียน ควรรู

Ç¸Ô Õ·Òí a = 3, b = 7 áÅÐ c = -6 ÊÁ¡ÒùéÕäÁ‹ÁÕ¤íҵͺ ËÁÒ¶֧ äÁ‹ÁÕ¨íҹǹ¨ÃÔ§ã´à»š¹¤íҵͺ¢Í§
¨Ò¡ x = -b ± 2ba2 - 4ac ÊÁ¡Òù¹Ñé æ
á·¹¤‹Ò x = -b ± 429(3-) 4(3)(6)
= -7 ±6 11
x = -3 ËÃ×Í 32 32
x

´Ñ§¹éѹ ¤íҵͺ¢Í§ÊÁ¡Òà ¤×Í -3 áÅÐ

µÍº¢ÍŒ 2.

91

นาํ Warm-up สอน Instruction สสรรปุปุ WWrraapp--uupp

สรปุ

15. ¼ÙŒÊ͹ÊÒ¸Ôµ¡ÒÃá¡ŒÊÁ¡ÒáíÒÅѧÊͧ แบบฝกทักษะท่ี 3.3
µÑÇá»Ãà´ÕÂÇâ´ÂÇÔ¸Õ¡ÒÃ㪌Êٵà µÒÁ·Õè
⨷¡íÒ˹´¨Ò¡áºº½ƒ¡·Ñ¡ÉзÕè 3.3 ค�ำชี้แจง ให้ผู้เรียนแสดงวิธีท�าเพ่ือหาค�าตอบของสมการก�าลงั สองตวั แปรเดยี วต่อไปน้ี โดยวธิ ี
ãËŒ¼ŒÙàÃÕ¹´Ù໹š µÑÇÍÂÒ‹ § การใชส้ ูตร พรอ้ มทัง้ ตรวจสอบคา� ตอบ

16. ¼ÙŒÊ͹ãËŒ¼ÙŒàÃÕ¹ᵋÅФ¹·íÒẺ½ƒ¡ 2
·Ñ¡ÉзÕè 3.3
1. x + 2x - 16 = 0 ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ�.. .....จ...ำ..ก...ส...ม....ก...ำ..ร.. ......x..2.. ..+.. ..2...x... ..-.. ..1..6... .....=.. .....0... ..........จ...ะ..ไ..ด...ว้..ำ.่ ...........................................................................................
เกร็ดแนะ ผสู อน
......... .............................................จa......... ำ......=ก...... ... ...1.........,... ......b......... ......=...x...... ... ...2...... ......=แ...... ...ล.........ะ......-... ......cb...... ... ...=...±... ...... -...... ...1... 2...6...b...a... ...2...... ......-...... ......4.........a.........c............................................................................................. (จ แ...3...ะท.......ไ...1น...ด...2...ค...ว้ 3......่ำำ่ ...1... ...)...x...2... ......+=...... ......2-......(1......3 ...1+.......61... ... ...2...-1...3... 7...1...1... )...6...= ...... - ......3 ...=1....... 1...6...0...2 ......=3...... 1......0 ......ใ......น.........ส.........ม......ก.........ำ......ร...... ..................
¼ÙŒÊ͹¤ÇÃÊÒ¸ÔµµÑÇÍ‹ҧ¡ÒÃá¡Œ ... ............... .....................x... .....=.. ......-..2... ..±... .. .. ..2...22.. .(.-.1. .).4...(..1..)..(..-..1...6...)................... แ..ท...น...ค...่ำ.. ..x.. ..=.. ..-..1.. .-.. .. ..1..7... .0=... ..-.=.5. ...10.. 2. .ส.3..ม1.. .ก.ใ.ำ.น.ร..สเ.ป.ม..น็.ก..จ.ำ.ร.ร.งิ. ....
ÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ãà´ÕÂÇâ´Â㪌ÊÙµÃ
ã˼Œ ŒÙàÃÂÕ ¹´Ù ¨Òí ¹Ç¹ 1 ¢ÍŒ â´ÂÊ‹ÁØ ¼àŒÙ ÃÂÕ ¹ ........................................................................................................................จ..ะ..ไ..ด...ว้ ..ำ่.......................................................................
ÁÒ 1 ¤¹ áÅŒÇãËŒ¼ÙŒàÃÕ¹àÅ×Í¡Ç‹Ò¨ÐãËŒ
¼ÙŒÊ͹ÊÒ¸Ôµ¡ÒÃá¡ŒÊÁ¡ÒáíÒÅѧÊͧ ... ............... .....................x... .....=.. ......-..2... ..±... .. ..2 ..4... ..+.. ..6...4..................................... ( .-..5.....1..2...3..1..)..2. ..+.. ..2..(.1.-.6.5. ...-1.. .2.1.3.6.1.. .). .=.-. . .0.1..6.. ..=.. ..0.....................
µÑÇá»Ãà´ÕÂÇâ´ÂãªÊŒ µÙ Ã㹢͌ ã´
........................................................................................................................ ........................................0... ...=.. ..0.. . .ส...ม...ก..ำ..ร..เ.ป...น็...จ..ร..งิ.....

x = -2 68..............................................................±..........................................................ด...งั ..น...้ัน.. ..ค...ำ่..ข...อ..ง.. ..x... .ค...อื... .-..1... .+.. .. .1...7.. .ห...ร...ือ.. ..-..1.. .-.. .. ..1..7.....
x = -1 2 17 .............................................................±.............................................................................................................................................
... ...............ด...ง.ั ..น...น้ั... .. ........ค...ำ�...ต...อ...บ...ข...อ...ง..ส...ม...ก...ำ...ร.. ..ค...อื... ..-...1.. ..+... .. .. ..1..7... ..ห...ร..อื... ..-...1.. ..-.. .. .. ..1...7... .............................................................

..........................................................................................................................................................................................................

2

2. x + 6x - 6 = 0 ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ�.. .....จ...ำ..ก...ส...ม....ก...ำ..ร.. ........x...2.. ..+.. ..6...x.. ..-.. ..6... .....=.. .....0... ..........จ...ะ..ไ..ด...ว้..ำ.่ ...........................................................................................

......... .............................................aจ......... ำ......=ก...... ... ...1.........,... ......b......... ......=...x...... ... ...6...... ......=แ...... ...ล.........ะ......-... ......cb...... ... ...=...±... ...... -...... ...6... 2... ...b...a......2...... ......-...... ......4.........a.........c............................................................................................. แจ( ...0...ะท.......ไ...8น...ด...7...ค...ว้ 2......่ำำ่ ...9... ...)...x...2... ......+=...... ......6-......(3......0 ...+.......8... ...6 ...7...1 ...2...-5...9... ... )...6...= ...... - ......0 ...=6....... 8... ...0...=7...... 2......09...... ......ใ......น.........ส.........ม......ก.........ำ......ร...... ..................
... ............... .....................x... .....=.. ......-..6... ..±... .. .. ..6..2.2.( ..1-...) ..4...(.1...)..(..-..6...)......................แ ..ท...น...ค...่ำ.. ..x.. ..=.. ..-..3.. .-.. .. ..1..5... .0=... ..-.=.6. ...80.. 7. .ส.2..ม9.. .ก.ใ.ำ.น.ร..สเ.ป.ม..น็.ก..จ.ำ.ร.ร.งิ. ....

........................................................................................................................จ..ะ..ไ..ด...ว้ ..่ำ.......................................................................

... ............... .....................x... .....=.. .....-...6.. ...±... .. .. .2.3...6.. ..+... ..2...4.................................. ( .-..6.....8..7...2..9..)..2. ..+.. ..6..(..-..6.6...8 ..-.7. .2.6.9.. .). .=.-. . .0.6.. ..=.. ..0.......................

........................................................................................................................ ........................................0... ...=.. ..0.. . .ส...ม...ก..ำ..ร..เ.ป...น็...จ..ร..งิ.....

x = -6 60..............................................................±..........................................................ด...งั ..น...น้ั .. ..ค...่ำ..ข...อ..ง.. ..x... .ค...อื... .-..3... .+.. .. .1...5.. .ห...ร...ือ.. ..-..3.. .-.. .. ..1..5.....
x = -3 2 15 .............................................................±.............................................................................................................................................
... ...............ด...ง.ั ..น...น้ั... .. ........ค...ำ�...ต...อ...บ...ข...อ...ง..ส...ม...ก...ำ...ร.. ..ค...อื... ..-...3.. ..+... .. ..1..5... ..ห...ร..อื... ..-...3.. ..-.. .. ..1...5... .................................................................

92

Vocational Skills ขV้อ-สNอบEแTนว

·¡Ñ ÉСÒÃá¡ÊŒ Á¡ÒáÒí Å§Ñ ÊͧµÇÑ á»Ãà´ÂÕ Çâ´ÂãªÊŒ µÙ à ໹š ʧèÔ ¨Òí ໹š ÊÁ¡ÒÃã¹¢ŒÍã´ÁÕ¤íҵͺà·Ò‹ ¡ºÑ 2 +3 5 áÅÐ 2 -3 5
ÊÒí ËÃºÑ ¹¡Ñ ʶµÔ ËÔ ÃÍ× ¹¡Ñ ¤³µÔ ÈÒʵÏ «§Öè 㹤ÇÒÁ໹š ¨Ã§Ô 㹡Òû¯ºÔ µÑ §Ô Ò¹ 1. 5x2 + 6x - 1 = 0
ÊÁ¡ÒÃÊÇ‹ ¹ãËÞ‹¨ÐÁÕ¤‹Ò໹š ¨Òí ¹Ç¹·äèÕ ÁÊ‹ ÒÁÒöᡵÑÇ»ÃСͺ䴌 2. x2 + 3x - 54 = 0
3. 9x2 - 12x - 1 = 0
4. 6x2 - 3x - 18 = 0
5. 25x2 - 15x + 36 = 0

92

นาํ Warm-up สอน Instruction สสรรปุปุ WWrraapp--uupp

3. 6x2 - 10x - 8 = 0 สรปุ
ตรวจสอบค�ำตอบ............ ว........ธิ............ที ............ำ�........ ....................aจจ............ ำำ........=กก........ .... ส....6........ม....,.... ........กb............ำ ........ร=....x.... ........ -....6........1....x....=0....2.... .... ........แ-........ ....ล-....1........bะ....0.... ....x....c±........ .... .... -....=.... .... .... 28....-....b.... a8........2.... ....=....-........ .... ........4........0....a........ c........................................จ........ะ............ไ........ด............ว้ ........ำ่.................................... แ จ6............(ะท....2....ไ....น.....ด....2....ค....ว้5....่ำ....ำ่ 7........ ....3x........).... ....2....= ........- ........ 5........1 ........0+........( ....6 ....2....8 .........7 2........-3....5.... ....7 ....8....=3.... .... )........2 =........-. ....2.... ....08....5........ 7....=....3........ .... ....0ใ........น............ส............ม........ก............ำ........ร........ ........................
17. ¼ÙŒÊ͹ÊØ‹ÁµÑÇá·¹¼ÙŒàÃÕ¹ÍÍ¡ÁÒáÊ´§
............... ........................................................................... .........................................................................................................xxx............... .........................===.......... .........................11.....1..........000............... ..........±±±..........1..... ..........2 .......... ..........12(1..........-.....092..........102.....2.....0..... .....(.....)+.....6.....2..... ) ..........1-..........9..... ..........42..........(..........6...............)..........(..........-..........8...............)............................................................ ด แ 6จ...............(ะทัง.....-.....ไน.....น0.....ด.....้ัน.....ค......้ว5..... ำ่.....ำ่9ค.......... .....0ำ่x..........7ข..... .....).....=อ.....2..... .....ง.....-5..... ..... x ..........1-..... .....ค0 .....6 .....(.....ือ8 .....-.....7 .....0.....-35........... 5 ..... .....800+.....9=..... ..... 6 0 ..........-=== 7..........07 .....)..... .3.....0005.....-..... .....9 ห.....สส.....80.....ร.....มม 7.....อื.....=กก..... ..... ใ.....ำำ.....05น.....รร..... .....เเส.....ปป-..........ม 6.....นน็็ ..........ก จจ..........7ำ.....รร.....3ร.....งงิิ .......... ............... ÇÔ¸Õ·íÒ¡ÒÃá¡ŒÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ã
à´ÕÂÇâ´ÂÇÔ¸¡Õ ÒÃãªÊŒ µÙ Ãã¹áºº½ƒ¡·¡Ñ ÉÐ
.......................................................................................................................................................................................................... º¹¡Ãдҹ

...... ..............................ด ......งั......น......นั้...... .... ..............x..ค.... ..ำ�....ต....=....อ ......บ....5..ข.... ..อ..±....ง.. ..6ส.. ....ม ....7..ก....3..ำ.... ..ร.... ....ค......อื...... ....5...... ....+...... ..6.. .... ....7......3..........ห......ร......อื .... ......5...... ....-.... ..6.. .... ....7......3.................................................................................................................................... 18. ¼ÙŒàÃÕ¹áÅмٌÊ͹ËÇÁ¡Ñ¹µÃǨÊͺ
¤ÇÒÁ¶Ù¡µŒÍ§¢Í§¡ÒÃáÊ´§ÇÔ¸Õ·íÒ¡ÒÃ
.......................................................................................................................................................................................................... á¡ŒÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ãà´ÕÂÇâ´Â
Ç¸Ô ¡Õ ÒÃãªÊŒ ÙµÃ
2
เกร็ดแนะ ผูสอน
4. 2x - 3x + 1 = 0 ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ�.. .....จ...ำ..ก...ส...ม....ก...ำ..ร.. ......2..x...2. ..-... ..3..x... ..+.. ..1... ....=... .....0... ..........จ..ะ...ไ..ด...ว้ ..ำ่............................................................................................
¼ÙŒÊ͹¤ÇÃແ´âÍ¡ÒÊãËŒ¼ÙŒàÃÕ¹䴌
................................. .....................................................................................................................................................................ดaจ ................................. ำงั......................=ก...........น........... ........... ...........น้ั2................................., ........... ........... ...........b................................. ......................=...........xxxxxx...........ค ...................... -...........ำ�......................3ต......................====== ......................อ แ.................................บล......................3333...........44ขะ-...................... ...........bอ ...........±±4+±...........c= ...........ง4........... ±...........ส...........1= 1...................... ม4 ...........,,........... 1 (1...........9ก........... 32-...........42 ...........bำ...........3 a2-......................ร24-)........... (= ........... 2 ...........28-........... ค1...........)........... 21-...........อื...........4 ...................... 4...........a...........1...........(c......................2......................ห...........)...........(...........ร...........1......................อื ...........)...................... ......................21................................................................................................................................................................................ แ แ ด.................................ททัง......................น...........นน......................ัน้...........คค2...........2...........2 ่ำ่ำ...........(...........(ค(...........21 1...........14xx...........ำ่)...........))...........2 ข2........... ........... ==อ...........---12........... ...........2 ง ...........1 123...........3 3 ...........- ...........-((x(........... ใใ...........2211 1น ...........3232น3...........)))ค........... ส........... ส...........-+ือ++++...........มม........... ........... ...........ก2322101110ก...........1...........ำ........... ำ......................หรร...........========........... ...........ร ...........จจ...........อื ...........00000000ะ...........ะ ........... ไ...........ไ12 ...........ด...........ดสส......................้ว้ว...........มม...........ำ่...........่ำกก.................................ำำ......................รร......................เเ...........ปป.................................นน็็.................................จจ......................รร.................................งงิิ ....................................................... «Ñ¡¶ÒÁ ¢³Ð·èÕ¡íÒÅѧ·íÒẺ½ƒ¡·Ñ¡ÉÐ
à¾×èÍ໚¹¡ÒÃà¾ÔèÁ¤ÇÒÁÃÙŒ¤ÇÒÁࢌÒã¨ãËŒ
.......................................................................................................................................................................................................... ¡ºÑ ¼àŒÙ ÃÕ¹

93

ÇÔà¤ÃÒÐˏ¤Òí µÍº

Ç¸Ô Õ·íÒ àÁ×Íè á¡ÊŒ Á¡Ò÷¡Ø µÑÇàÅÍ× ¡¨Ð¾ºÇ‹Ò µÑÇàÅ×Í¡·Õè 3
Á¤Õ íҵͺà·Ò‹ ¡Ñº 2 +3 5 áÅÐ 2 -3 5
-(-12) ± 124(94) - 4(9)(-1)
á·¹¤Ò‹ x = 2 ±3 5
x =

µÍº¢ÍŒ 3.

93

นาํ Warm-up สอน Instruction สสรรปุปุ WWrraapp--uupp
สรปุ
5. 23 - x - 10x2 = 0
19. ¼ÊŒÙ ͹ÊÁ‹Ø ¼àŒÙ ÃÂÕ ¹ÍÍ¡ÁÒáÊ´§Ç¸Ô ·Õ Òí â´Â ตรวจสอบค�ำตอบ.........ว ......ธิ.........ที .........ำ� ...... ...............จaจ......... ำำ......=กก...... ...ส ...-......ม...1......0ก.........,ำ... ......รb...x... ... ... ...=2...... ...3=...-... ... ...1-...... ... ...แx...-......ล b......-...ะ ... ...± ...1...c...... 0... ... x...=... 2...2 ...b... a2......2...3 ...=... -...... ... ......4......0a......... c..............................จ......ะ.........ไ......ด.........ว้ ......ำ่...........................จ2 แ.........3ทะ...... ไ...น...-ด...ค... ...้ว...(ำ่...่ำ-... ...x1...... ....=...5... ...6-...... ...7(......41......) ...... +......- 2... ... ...0 1......9...0...2...(...1...-......)1... ...=....5... ...-...6...1...7.......45.........)6...2...7... ...4=...... ... ใ...0...น.........ส......ม.........ก.........ำ......ร...... .........
¼ÙŒÊ͹áÅмٌàÃÕ¹¤¹Í×è¹æ ¨Ðª‹Ç¡ѹ
µÃǨÊͺ¤íҵͺ¢³Ð·ÕèµÑÇ᷹͸ԺÒ ....................................................................................................................... .......2...4....5..6...7..4.. ..-.. ..2..4.....5..6...7..4.. ...=... ..0................................
˹ŒÒª¹Ñé àÃÂÕ ¹ ËÒ¡¾ºÇÒ‹ µÑÇá·¹¼ÙŒàÃÕ¹
à©ÅÂäÁ¶‹ ¡Ù µÍŒ § ¼ÊŒÙ ͹áÅмàŒÙ ÃÂÕ ¹ªÇ‹ ¡¹Ñ ........................ ........................................................................................................................ ........................................................................................................................................................................xxxx........................ ................................====........................ ........................................-1................1 ........ (........1 ........±1........±........ ........ ........-±........ + ........2........ ........ 2........ 0 ........( ........90- -........ ........122........91........2........ 102)................+(2........1........ -........ ........)-1........9,........ ........02................4-........)0........(........(........-1................1........ ................0-........................ )2................( ........02........ ................93................2........)................1........................)........................................................ แ ด 2จ........................3ทะงั................ ไน........น........-ด........้ันค........ ........้ว........1 ำ่........ำ่ ........ค. ห........4x................ำ่ร........6 ........=ขือ........7........ ........อ 4-................-ง ........(........ - ........(1x........ ........1 2........ 1........ -ค........30........- 2........ ือ........( 2........-0 1........ 0........ 9 .........-9........224........ ........2(30016........1........1 ........7)........ )........4====........+........) ........ 2........1 2........000 0........ .........=4 9........ ........สส 6........2........0........7มม1................4กก)................ ........ำำใ................นรร................เเ........สปป................ม........นน็็ ................กจจ........................ำรร................รงงิิ ................ ........................
á¡äŒ ¢ã˶Œ ¡Ù µÍŒ § ... ...............ด...งั...น...น้ั ... .. ........ค...ำ� ..ต....อ...บ...ข...อ...ง..ส...ม...ก...ำ...ร.. ..ค...อื... ..-.. .(..1... ..+.. .2.. .0. ..9..2...1...)....ห...ร..อื... ..-... .(.1... ..-... 2.. .0. ..9..2...1...)...............................................

เกร็ดแนะ ผูสอน 6. x2 + 6x = -2
ตรวจสอบค�ำตอบ...ว..ธิ...ที ...ำ�.. .....จ...ำ...ก..ส....ม...ก...ำ..ร... ................x...2. ..+... ..6..x... .....=.. .....-...2..................................................................................................................
¼ÙŒÊ͹¤ÇÃÊÃØ»ÊÒÃÐÊíÒ¤ÑÞãˌᡋ
¼ÙŒàÃÕ¹¡‹Í¹¨Ð¡Å‹ÒǶ֧ËÑÇ¢ŒÍ⨷»˜ÞËÒ ..................... ......................................................................................................... aจ ..................... ำ..............=.......ก.............. ..............1..............,.............. ..............b.............. ..............=.......xxx.............. ....... .......6.............. ..............===แ.......x....... .......ล.......2....... .......ะ.......+.......---....... ....... 66.......cb.......6....... ....... x.......±=±.......±....... ....... ....... +2....... ....... ....... 2 ....... 2.......263.......2b....... a.......26(.......2....... 1....... .......-=.......--)....... ....... .......4.......48..............(0.......a.......1....... .......c)..............(..............2.....................)..............จ.....................ะ..............ไ..............ด.....................ว้..............ำ.......่ ............................................................... แ(จ แจ (.......--.......ะะทท.......50.......ไไ.......นน.........ดด.......63.......คค.......้ว้ว45..............่ำำ่ำ่่ำ3.......54....... .......382.......xx.......0.)).......8 .......22........== 17..............++ .......25.......-- .......5 .......6633.......- .......(( ....... --+.......--.......3 ....... 05.......03 ................77..36..............18 .......45.......27==.......54.......5500 ....... 82.......--....... .......))50....... .......====.........++....... 6 3....... .......400005.......22....... 5.......4 .......สส.......8==2..............มม ..............ใ00ใ.......กก.......นน.......ำำ.......ส.......สรร..............มเเม.......ปป..............กก.......นน็็..............ำำ.......จจ.......รร.......รร....... .......งงิิ...................................
¢Í§ÊÁ¡ÒáíÒÅѧÊͧµÑÇá»Ãà´ÕÂÇ ¾ÃŒÍÁ
¡µÇÑ ÍÂÒ‹ §»ÃСͺà¾ÁÔè àµÁÔ áÅСÒõÃǨ ........................................................................................................................ด...ัง..น...ั้น.. ..ค...ำ่..ข...อ..ง... .x... ..ค...อื .. ..-..3.. .+.. .. ..7.. ..ห...ร..อื... ..-..3.. .-.. .. ..7........
Ẻ½¡ƒ ·¡Ñ ÉТͧ¼àŒÙ ÃÂÕ ¹ ËÒ¡¼àŒÙ ÃÂÕ ¹¤¹ã´
ä´·Œ íÒẺ½ƒ¡·¡Ñ Éм´Ô ËÅ¡Ñ ¡Òà ãËŒ¼ÊÙŒ ͹ x = -6 28..............................................................±............................................................................................................................................
͸ԺÒÂãË¡Œ ºÑ ¼ÙŒàÃÕ¹໚¹ÃÒº¤Ø ¤Å x = -3 2 7 .............................................................±.............................................................................................................................................
... ...............ด....งั ..น...น้ั... .. ........ค...ำ�...ต...อ...บ...ข...อ...ง..ส...ม...ก...ำ...ร.. ..ค...อ.ื .. ..-...3.. ..+... .. ..7.. ..ห...ร...อื... ..-..3... ..-.. .. ..7.........................................................................

.........................................................................................................................................................................................................

94

ขVอ้ -สNอบEแTนว

¤Òí µÍº¢Í§ÊÁ¡Òà x2 + 6x - 2 = 0 µÃ§¡ºÑ ¢ÍŒ ã´
1. 5 ± 17
2. 8 ± 21
3. -6 ± 31
4. -3 ± 11
5. -1 ± 71

94