Question

factor quadratic expressions
factor each quadratic expression.

1. $x^2 - 11x + 30$
2. $x^2 - x - 30$
3. $-x^2 - x + 30$
4. $9x^2 + 24x + 16$
5. $81 - 4x^2$
6. $2x^2 - 4x - 16$

Explanation:

Step1: Find pair for sum -11, product 30

Look for two numbers that add to $-11$ and multiply to $30$: $-5$ and $-6$.
$x^2 -11x +30=(x-5)(x-6)$

Step2: Find pair for sum -1, product -30

Look for two numbers that add to $-1$ and multiply to $-30$: $5$ and $-6$.
$x^2 -x -30=(x+5)(x-6)$

Step3: Factor out -1 first

$-x^2 -x +30=-(x^2 +x -30)$
Find pair for sum 1, product -30: $6$ and $-5$.
$-(x^2 +x -30)=-(x+6)(x-5)$

Step4: Recognize perfect square trinomial

$9x^2=(3x)^2$, $16=4^2$, $24x=2\times3x\times4$.
$9x^2 +24x +16=(3x+4)^2$

Step5: Recognize difference of squares

$81=9^2$, $4x^2=(2x)^2$. Use $a^2-b^2=(a-b)(a+b)$.
$81-4x^2=(9-2x)(9+2x)$

Step6: Factor out GCF first

$2x^2 -4x -16=2(x^2 -2x -8)$
Find pair for sum -2, product -8: $2$ and $-4$.
$2(x^2 -2x -8)=2(x+2)(x-4)$

Answer:

1. $(x-5)(x-6)$
2. $(x+5)(x-6)$
3. $-(x+6)(x-5)$
4. $(3x+4)^2$
5. $(9-2x)(9+2x)$
6. $2(x+2)(x-4)$