# Factor the following quadratics. a) 2x^{2}+7x+6 b) 4x^{2}-13x+9 c) 3x^{2}-10x+8 d) 2x^{2}-3x-2 e) 3x^{2}+17x+20 f) 4x^{2}-25x+25

a) $2{x}^{2}+7x+6$
b) $4{x}^{2}-13x+9$
c) $3{x}^{2}-10x+8$
d) $2{x}^{2}-3x-2$
e) $3{x}^{2}+17x+20$
f) $4{x}^{2}-25x+25$
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Jimmy Macias
Step 1
a) $2{x}^{2}+7x+6$
Factor the quadratic $2{x}^{2}+7x+6$
The coefficient of ${x}^{2}$ is 2 and the constant term 6
The product of 2 and 6 is 12
The factor of 12 which sum to 7 ae 3 and 4
So $2{x}^{2}+7x+6=2{x}^{2}+4x+3x+6$
$=2x\left(x+2\right)+3\left(x+2\right)$
$=\left(2x+3\right)\left(x+2\right)$
$⇒2{x}^{2}+7x+6=\left(2x+3\right)\left(x+2\right)$
Step 2
$4{x}^{2}-13x+9$
The coefficient of ${x}^{2}$ is 4 and the constant term is 9. The product of 4 and 9 is 36
The factors of 36 which sum to -13 are -4 and -9
So $4{x}^{2}-13x+9=4{x}^{2}-9x-4x+9$
$=x\left(4x-9\right)-1\left(4x-9\right)$
$=\left(4x-1\right)\left(4x-9\right)$
$⇒4{x}^{2}-13x+9=\left(4x-1\right)\left(4x-9\right)$
Step 3
$3{x}^{2}-10x+8$
The coefficient of ${x}^{2}$ is 3 and the constant term is 8
The product o 3 and 8 is 24
The factors a 24 which sm to -10 are -4 and -6
So $3{x}^{2}-10x+8=3{x}^{2}-6x-4x+8$
$=3x\left(x-2\right)-4\left(x-2\right)$
$=\left(3x-4\right)\left(x-2\right)$
$⇒3{x}^{2}-10x+8=\left(3x-4\right)\left(x-2\right)$
Step 4
$2{x}^{2}-3x-2$
The coefficien of ${x}^{2}$ is 2 and the constant term is -2
The product of 2 and -2 is -4
The factors of -4 which sum to -3 are 1 and -4
So $2{x}^{2}-3x-2=2{x}^{2}-4x+x-2$
$=2x\left(x-2\right)+1\left(x-2\right)$
$=\left(2x+1\right)\left(x-2\right)$
$⇒2{x}^{2}-3x-2=\left(2x+1\right)\left(x-2\right)$
Step 5
$3{x}^{2}+17x+20$
The coefficient of ${x}^{2}$ is 3 and the constant is 20
The product of 3 and 20 is 60
The factors of 60 which sum to 17 are 5 and 12
So
aquariump9
Step 1
a) $2{x}^{2}+7x+6$
$a+b=7$
$ab=2×6=12$ NKS

$1+12=13$
$2+6=8$
$3+4=7$
$a=3$
$b=4$
Rewrite
$\left(2{x}^{2}+3x\right)+\left(4x+6\right)$
$x\left(2x+3\right)+2\left(2x+3\right)$
$\left(2x+3\right)\left(x+2\right)$
b) $4{x}^{2}-13x+9$
$a+b=-13$
$ab=4×9=36$

$-1-36=-37$
$-2-18=-20$
$-3-12=-15$
$-4-9=-13$
$-6-6=-12$
$a=-9$
$b=-4$
Rewrite
$\left(4{x}^{2}-9x\right)+\left(-4x+9\right)$
$x\left(4x-9\right)-\left(4x-9\right)$
$\left(4x-9\right)\left(x-1\right)$
Step 2
c) $3{x}^{2}-10x+8$
$a+b=-10$
$ab=3×8=24$

$-1-24=-25$
$-2-12=-14$
$-3-8=-11$
$-4-6=-10$
$a=-6$
$b=-4$
Rewrite
$\left(3{x}^{2}-6x\right)+\left(-4x+8\right)$
$3x\left(x-2\right)-4\left(x-2\right)$