មេរៀន៖ កន្សោមពីជគណិត - ថ្នាក់ទី៩ (Algebraic Expressions - Grade 9)
(Full Step-by-Step Answer Key)
ភាគ ៤៖ វិធីបំពេញបន្ថយតួ និងញែកតួកណ្តាល (Part 4: Completing the Square & Splitting Middle Term)
រៀបរៀងដោយ៖ លោកគ្រូ CHEANG SOKKONG
💡 គន្លឹះ (Tip): នៅក្នុងភាគនេះ យើងនឹងសិក្សាពីការផ្ដុំតួ ៣ ដើម្បីបង្កើតជាការ៉េនៃទ្វេធា $(a \pm b)^2$ ការដាក់ជាផលគុណកត្តាតាម **វិធីបំពេញនិងបន្ថយតួ** និង **វិធីញែកតួកណ្តាល**។
👉 បើកន្សោមវែងពេក ប្អូនៗអាចអូសទៅឆ្វេង-ស្ដាំបាន។
២៣. ការផ្ដុំតួជាការ៉េនៃទ្វេធា រួចប្រើផលសងការ៉េ
ក. $a^2+2ab+b^2-9c^2$
$= (a^2+2ab+b^2) - 9c^2$
$= (a+b)^2 - (3c)^2$
$= (a+b-3c)(a+b+3c)$
$= (a+b)^2 - (3c)^2$
$= (a+b-3c)(a+b+3c)$
ខ. $x^2+y^2-z^2-2xy$
$= (x^2-2xy+y^2) - z^2$
$= (x-y)^2 - z^2$
$= (x-y-z)(x-y+z)$
$= (x-y)^2 - z^2$
$= (x-y-z)(x-y+z)$
គ. $25x^2-10x+1-36y^2$
$= [(5x)^2-2(5x)(1)+1^2] - (6y)^2$
$= (5x-1)^2 - (6y)^2$
$= (5x-1-6y)(5x-1+6y)$
$= (5x-1)^2 - (6y)^2$
$= (5x-1-6y)(5x-1+6y)$
ឃ. $9a^2+6a+1-36b^2$
$= [(3a)^2+2(3a)(1)+1^2] - (6b)^2$
$= (3a+1)^2 - (6b)^2$
$= (3a+1-6b)(3a+1+6b)$
$= (3a+1)^2 - (6b)^2$
$= (3a+1-6b)(3a+1+6b)$
ង. $9-a^2+2ab-b^2$
$= 9 - (a^2-2ab+b^2)$
$= 3^2 - (a-b)^2$
$= [3-(a-b)][3+(a-b)]$
$= (3-a+b)(3+a-b)$
$= 3^2 - (a-b)^2$
$= [3-(a-b)][3+(a-b)]$
$= (3-a+b)(3+a-b)$
ច. $x^2-y^2+6y-9$
$= x^2 - (y^2-6y+9)$
$= x^2 - (y-3)^2$
$= [x-(y-3)][x+(y-3)]$
$= (x-y+3)(x+y-3)$
$= x^2 - (y-3)^2$
$= [x-(y-3)][x+(y-3)]$
$= (x-y+3)(x+y-3)$
ឆ. $1+2ab-(a^2+b^2)$
$= 1+2ab-a^2-b^2$
$= 1 - (a^2-2ab+b^2)$
$= 1^2 - (a-b)^2$
$= [1-(a-b)][1+(a-b)]$
$= (1-a+b)(1+a-b)$
$= 1 - (a^2-2ab+b^2)$
$= 1^2 - (a-b)^2$
$= [1-(a-b)][1+(a-b)]$
$= (1-a+b)(1+a-b)$
ជ. $a^2-b^2-4ac+4c^2$
$= (a^2-4ac+4c^2) - b^2$
$= (a-2c)^2 - b^2$
$= (a-2c-b)(a-2c+b)$
$= (a-2c)^2 - b^2$
$= (a-2c-b)(a-2c+b)$
ឈ. $4a^2-4b^2+4a+1$
$= (4a^2+4a+1) - 4b^2$
$= (2a+1)^2 - (2b)^2$
$= (2a+1-2b)(2a+1+2b)$
$= (2a+1)^2 - (2b)^2$
$= (2a+1-2b)(2a+1+2b)$
២៤. សរសេរកន្សោមជាទម្រង់ $(x+a)^2+b$
ក. $x^2+6x+4$
$= x^2 + 2(x)(3) + 3^2 - 3^2 + 4$
$= (x+3)^2 - 9 + 4$
$= (x+3)^2 - 5$
$= (x+3)^2 - 9 + 4$
$= (x+3)^2 - 5$
ខ. $x^2-4x+7$
$= x^2 - 2(x)(2) + 2^2 - 2^2 + 7$
$= (x-2)^2 - 4 + 7$
$= (x-2)^2 + 3$
$= (x-2)^2 - 4 + 7$
$= (x-2)^2 + 3$
គ. $x^2+14x+44$
$= x^2 + 2(x)(7) + 7^2 - 7^2 + 44$
$= (x+7)^2 - 49 + 44$
$= (x+7)^2 - 5$
$= (x+7)^2 - 49 + 44$
$= (x+7)^2 - 5$
ឃ. $x^2-12x+30$
$= x^2 - 2(x)(6) + 6^2 - 6^2 + 30$
$= (x-6)^2 - 36 + 30$
$= (x-6)^2 - 6$
$= (x-6)^2 - 36 + 30$
$= (x-6)^2 - 6$
ង. $x^2+10x+17$
$= x^2 + 2(x)(5) + 5^2 - 5^2 + 17$
$= (x+5)^2 - 25 + 17$
$= (x+5)^2 - 8$
$= (x+5)^2 - 25 + 17$
$= (x+5)^2 - 8$
ច. $x^2+22x+141$
$= (x^2+22x+121) - 121 + 141$
$= (x+11)^2 + 20$
$= (x+11)^2 + 20$
ឆ. $x^2+24x+121$
$= x^2+2(x)(12)+12^2$
$= (x+12)^2 - 144 + 121$
$= (x+12)^2 - 23$
$= (x+12)^2 - 144 + 121$
$= (x+12)^2 - 23$
ជ. $x^2-16x+57$
$= (x^2-16x+64) - 64 + 57$
$= (x-8)^2 - 7$
$= (x-8)^2 - 7$
ឈ. $x^2+18x+93$
$= (x^2+18x+81) - 81 + 93$
$= (x+9)^2 + 12$
$= (x+9)^2 + 12$
ញ. $x^2-2x+10$
$= (x^2-2x+1) - 1 + 10$
$= (x-1)^2 + 9$
$= (x-1)^2 + 9$
ដ. $x^2-8x-5$
$= (x^2-8x+16) - 16 - 5$
$= (x-4)^2 - 21$
$= (x-4)^2 - 21$
ឋ. $x^2+20x+83$
$= (x^2+20x+100) - 100 + 83$
$= (x+10)^2 - 17$
$= (x+10)^2 - 17$
២៥. ការដាក់ជាផលគុណកត្តាតាមវិធីបំពេញនិងបន្ថយតួ
ក. $x^2+6x+8$
$= x^2 + 2(x)(3) + 3^2 - 3^2 + 8$
$= (x+3)^2 - 9 + 8$
$= (x+3)^2 - 1^2$
$= (x+3-1)(x+3+1)$
$= (x+2)(x+4)$
$= (x+3)^2 - 9 + 8$
$= (x+3)^2 - 1^2$
$= (x+3-1)(x+3+1)$
$= (x+2)(x+4)$
ខ. $x^2-4x-12$
$= x^2 - 2(x)(2) + 2^2 - 2^2 - 12$
$= (x-2)^2 - 4 - 12$
$= (x-2)^2 - 4^2$
$= (x-2-4)(x-2+4)$
$= (x-6)(x+2)$
$= (x-2)^2 - 4 - 12$
$= (x-2)^2 - 4^2$
$= (x-2-4)(x-2+4)$
$= (x-6)(x+2)$
គ. $x^2-4x+3$
$= x^2 - 2(x)(2) + 2^2 - 2^2 + 3$
$= (x-2)^2 - 4 + 3$
$= (x-2)^2 - 1^2$
$= (x-2-1)(x-2+1)$
$= (x-3)(x-1)$
$= (x-2)^2 - 4 + 3$
$= (x-2)^2 - 1^2$
$= (x-2-1)(x-2+1)$
$= (x-3)(x-1)$
ឃ. $x^2+8x+15$
$= x^2 + 2(x)(4) + 4^2 - 4^2 + 15$
$= (x+4)^2 - 16 + 15$
$= (x+4)^2 - 1^2$
$= (x+4-1)(x+4+1)$
$= (x+3)(x+5)$
$= (x+4)^2 - 16 + 15$
$= (x+4)^2 - 1^2$
$= (x+4-1)(x+4+1)$
$= (x+3)(x+5)$
ង. $x^2-8x+15$
$= x^2 - 2(x)(4) + 4^2 - 4^2 + 15$
$= (x-4)^2 - 16 + 15$
$= (x-4)^2 - 1^2$
$= (x-4-1)(x-4+1)$
$= (x-5)(x-3)$
$= (x-4)^2 - 16 + 15$
$= (x-4)^2 - 1^2$
$= (x-4-1)(x-4+1)$
$= (x-5)(x-3)$
ច. $x^2-4x-32$
$= (x^2-4x+4) - 4 - 32$
$= (x-2)^2 - 36$
$= (x-2)^2 - 6^2$
$= (x-2-6)(x-2+6)$
$= (x-8)(x+4)$
$= (x-2)^2 - 36$
$= (x-2)^2 - 6^2$
$= (x-2-6)(x-2+6)$
$= (x-8)(x+4)$
ឆ. $x^2+x-12$
$= x^2 + 2(x)(\frac{1}{2}) + (\frac{1}{2})^2 - (\frac{1}{2})^2 - 12$
$= (x+\frac{1}{2})^2 - \frac{1}{4} - \frac{48}{4}$
$= (x+\frac{1}{2})^2 - \frac{49}{4}$
$= (x+\frac{1}{2})^2 - (\frac{7}{2})^2$
$= (x+\frac{1}{2}-\frac{7}{2})(x+\frac{1}{2}+\frac{7}{2})$
$= (x-\frac{6}{2})(x+\frac{8}{2})$
$= (x-3)(x+4)$
$= (x+\frac{1}{2})^2 - \frac{1}{4} - \frac{48}{4}$
$= (x+\frac{1}{2})^2 - \frac{49}{4}$
$= (x+\frac{1}{2})^2 - (\frac{7}{2})^2$
$= (x+\frac{1}{2}-\frac{7}{2})(x+\frac{1}{2}+\frac{7}{2})$
$= (x-\frac{6}{2})(x+\frac{8}{2})$
$= (x-3)(x+4)$
ជ. $x^2+7x+12$
$= x^2 + 2(x)(\frac{7}{2}) + (\frac{7}{2})^2 - (\frac{7}{2})^2 + 12$
$= (x+\frac{7}{2})^2 - \frac{49}{4} + \frac{48}{4}$
$= (x+\frac{7}{2})^2 - \frac{1}{4}$
$= (x+\frac{7}{2})^2 - (\frac{1}{2})^2$
$= (x+\frac{7}{2}-\frac{1}{2})(x+\frac{7}{2}+\frac{1}{2})$
$= (x+\frac{6}{2})(x+\frac{8}{2})$
$= (x+3)(x+4)$
$= (x+\frac{7}{2})^2 - \frac{49}{4} + \frac{48}{4}$
$= (x+\frac{7}{2})^2 - \frac{1}{4}$
$= (x+\frac{7}{2})^2 - (\frac{1}{2})^2$
$= (x+\frac{7}{2}-\frac{1}{2})(x+\frac{7}{2}+\frac{1}{2})$
$= (x+\frac{6}{2})(x+\frac{8}{2})$
$= (x+3)(x+4)$
ឈ. $x^2-8x+15$ (លំហាត់នេះដូចគ្នានឹងលំហាត់ ង)
$= (x-5)(x-3)$
ញ. $x^2-2x-48$
$= (x^2-2x+1) - 1 - 48$
$= (x-1)^2 - 49$
$= (x-1)^2 - 7^2$
$= (x-1-7)(x-1+7)$
$= (x-8)(x+6)$
$= (x-1)^2 - 49$
$= (x-1)^2 - 7^2$
$= (x-1-7)(x-1+7)$
$= (x-8)(x+6)$
ដ. $x^2+3x-10$
$= x^2 + 2(x)(\frac{3}{2}) + (\frac{3}{2})^2 - (\frac{3}{2})^2 - 10$
$= (x+\frac{3}{2})^2 - \frac{9}{4} - \frac{40}{4}$
$= (x+\frac{3}{2})^2 - \frac{49}{4}$
$= (x+\frac{3}{2})^2 - (\frac{7}{2})^2$
$= (x+\frac{3}{2}-\frac{7}{2})(x+\frac{3}{2}+\frac{7}{2})$
$= (x-\frac{4}{2})(x+\frac{10}{2})$
$= (x-2)(x+5)$
$= (x+\frac{3}{2})^2 - \frac{9}{4} - \frac{40}{4}$
$= (x+\frac{3}{2})^2 - \frac{49}{4}$
$= (x+\frac{3}{2})^2 - (\frac{7}{2})^2$
$= (x+\frac{3}{2}-\frac{7}{2})(x+\frac{3}{2}+\frac{7}{2})$
$= (x-\frac{4}{2})(x+\frac{10}{2})$
$= (x-2)(x+5)$
ឋ. $x^2+3x-54$
$= x^2 + 2(x)(\frac{3}{2}) + (\frac{3}{2})^2 - (\frac{3}{2})^2 - 54$
$= (x+\frac{3}{2})^2 - \frac{9}{4} - \frac{216}{4}$
$= (x+\frac{3}{2})^2 - \frac{225}{4}$
$= (x+\frac{3}{2})^2 - (\frac{15}{2})^2$
$= (x+\frac{3}{2}-\frac{15}{2})(x+\frac{3}{2}+\frac{15}{2})$
$= (x-\frac{12}{2})(x+\frac{18}{2})$
$= (x-6)(x+9)$
$= (x+\frac{3}{2})^2 - \frac{9}{4} - \frac{216}{4}$
$= (x+\frac{3}{2})^2 - \frac{225}{4}$
$= (x+\frac{3}{2})^2 - (\frac{15}{2})^2$
$= (x+\frac{3}{2}-\frac{15}{2})(x+\frac{3}{2}+\frac{15}{2})$
$= (x-\frac{12}{2})(x+\frac{18}{2})$
$= (x-6)(x+9)$
២៦. ការដាក់ជាផលគុណកត្តាតាមវិធីញែកតួកណ្តាល
ក. $2x^2+5x+3$
(តួកណ្តាល $5x$ ញែកជា $2x+3x$)
$= 2x^2 + 2x + 3x + 3$
$= 2x(x+1) + 3(x+1)$
$= (x+1)(2x+3)$
$= 2x^2 + 2x + 3x + 3$
$= 2x(x+1) + 3(x+1)$
$= (x+1)(2x+3)$
ខ. $2x^2+7x+5$
(តួកណ្តាល $7x$ ញែកជា $2x+5x$)
$= 2x^2 + 2x + 5x + 5$
$= 2x(x+1) + 5(x+1)$
$= (x+1)(2x+5)$
$= 2x^2 + 2x + 5x + 5$
$= 2x(x+1) + 5(x+1)$
$= (x+1)(2x+5)$
គ. $7x^2+9x+2$
(តួកណ្តាល $9x$ ញែកជា $7x+2x$)
$= 7x^2 + 7x + 2x + 2$
$= 7x(x+1) + 2(x+1)$
$= (x+1)(7x+2)$
$= 7x^2 + 7x + 2x + 2$
$= 7x(x+1) + 2(x+1)$
$= (x+1)(7x+2)$
២៧. ដាក់កន្សោមខាងក្រោមជាផលគុណកត្តា
ក. $2x^2-7x+5$
(តួកណ្តាល $-7x$ ញែកជា $-2x-5x$)
$= 2x^2 - 2x - 5x + 5$
$= 2x(x-1) - 5(x-1)$
$= (x-1)(2x-5)$
$= 2x^2 - 2x - 5x + 5$
$= 2x(x-1) - 5(x-1)$
$= (x-1)(2x-5)$
ខ. $7x^2-9x+2$
(តួកណ្តាល $-9x$ ញែកជា $-7x-2x$)
$= 7x^2 - 7x - 2x + 2$
$= 7x(x-1) - 2(x-1)$
$= (x-1)(7x-2)$
$= 7x^2 - 7x - 2x + 2$
$= 7x(x-1) - 2(x-1)$
$= (x-1)(7x-2)$
គ. $3x^2+5x-2$
(តួកណ្តាល $5x$ ញែកជា $6x-x$)
$= 3x^2 + 6x - x - 2$
$= 3x(x+2) - 1(x+2)$
$= (x+2)(3x-1)$
$= 3x^2 + 6x - x - 2$
$= 3x(x+2) - 1(x+2)$
$= (x+2)(3x-1)$
២៨. ដាក់កន្សោមខាងក្រោមជាផលគុណកត្តា
ក. $3x^2-5x-2$
(តួកណ្តាល $-5x$ ញែកជា $-6x+x$)
$= 3x^2 - 6x + x - 2$
$= 3x(x-2) + 1(x-2)$
$= (x-2)(3x+1)$
$= 3x^2 - 6x + x - 2$
$= 3x(x-2) + 1(x-2)$
$= (x-2)(3x+1)$
ខ. $3x^2+20x-7$
(តួកណ្តាល $20x$ ញែកជា $21x-x$)
$= 3x^2 + 21x - x - 7$
$= 3x(x+7) - 1(x+7)$
$= (x+7)(3x-1)$
$= 3x^2 + 21x - x - 7$
$= 3x(x+7) - 1(x+7)$
$= (x+7)(3x-1)$
គ. $2x^2-9x-5$
(តួកណ្តាល $-9x$ ញែកជា $-10x+x$)
$= 2x^2 - 10x + x - 5$
$= 2x(x-5) + 1(x-5)$
$= (x-5)(2x+1)$
$= 2x^2 - 10x + x - 5$
$= 2x(x-5) + 1(x-5)$
$= (x-5)(2x+1)$
២៩. ដាក់កន្សោមខាងក្រោមជាផលគុណកត្តា
ក. $6a^2+13a+6$
(តួកណ្តាល $13a$ ញែកជា $9a+4a$)
$= 6a^2 + 9a + 4a + 6$
$= 3a(2a+3) + 2(2a+3)$
$= (2a+3)(3a+2)$
$= 6a^2 + 9a + 4a + 6$
$= 3a(2a+3) + 2(2a+3)$
$= (2a+3)(3a+2)$
ខ. $2x^2+17x+21$
(តួកណ្តាល $17x$ ញែកជា $14x+3x$)
$= 2x^2 + 14x + 3x + 21$
$= 2x(x+7) + 3(x+7)$
$= (x+7)(2x+3)$
$= 2x^2 + 14x + 3x + 21$
$= 2x(x+7) + 3(x+7)$
$= (x+7)(2x+3)$
គ. $4x^2+13x+3$
(តួកណ្តាល $13x$ ញែកជា $12x+x$)
$= 4x^2 + 12x + x + 3$
$= 4x(x+3) + 1(x+3)$
$= (x+3)(4x+1)$
$= 4x^2 + 12x + x + 3$
$= 4x(x+3) + 1(x+3)$
$= (x+3)(4x+1)$
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