Before you read

Answer first — see what you already know.

A rectangular garden has area $x^{2} + 9 x + 20$ square feet. If the length and width are both expressions in $x$ , which factored form represents the dimensions?

A
$(x + 4) (x + 5)$
B
$(x + 2) (x + 10)$
C
$(x + 10) (x - 2)$
D
$(x + 20) (x + 1)$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

The cross-sectional area of a beam is modeled by $x^{2} - 8 x + 15$ . Which is the factored form?

A
$(x - 3) (x - 5)$
B
$(x + 3) (x - 5)$
C
$(x - 3) (x + 5)$
D
$(x + 3) (x + 5)$

Check your answer

Answer:

Theory

The Product-Sum Method

To factor $x^2 + bx + c$ , find two numbers $p$ and $q$ such that:

$p \times q = c \qquad \text{(product)}$
$p + q = b \qquad \text{(sum)}$

Then: $x^2 + bx + c = (x + p)(x + q)$

$x^2 + 5x + 6$
$(x + 2)(x + 3)$ find factors of 6 that sum to 5

Factoring a simple quadratic

It's like a puzzle: what two numbers multiply to $c$ and add to $b$ ?

y = x² + 5x + 6: roots at x = -3 and x = -2

Example 1

Factor $x^2 + 7x + 12$

Need: $p \times q = 12$ and $p + q = 7$ .

Pairs that multiply to $12$ : $(1,12), (2,6), (3,4)$ .

$3 + 4 = 7$ ✓

$x^2 + 7x + 12 = (x + 3)(x + 4)$

(x + 3)(x + 4)

Example 2

Factor $x^2 - 5x + 6$

Need: $p \times q = 6$ (positive) and $p + q = -5$ (negative).

Both numbers must be negative (negative sum, positive product).

$(-2) \times (-3) = 6$ ✓ and $(-2) + (-3) = -5$ ✓

$x^2 - 5x + 6 = (x - 2)(x - 3)$

(x - 2)(x - 3)

Example 3

Factor $x^2 + 2x - 15$

Need: $p \times q = -15$ (negative!) and $p + q = 2$ .

One number positive, one negative (since product is negative).

$5 \times (-3) = -15$ ✓ and $5 + (-3) = 2$ ✓

$x^2 + 2x - 15 = (x + 5)(x - 3)$

(x + 5)(x - 3)

Theory

Sign Patterns

The signs in $x^2 + bx + c$ tell you the signs of $p$ and $q$ :

$b$ (sum)	$c$ (product)	Signs of $p$ and $q$
$+$	$+$	Both positive
$-$	$+$	Both negative
$+$	$-$	One positive, one negative (larger is positive)
$-$	$-$	One positive, one negative (larger is negative)

This narrows down your search quickly.

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2 more sections including examples, practice problems, and step-by-step solutions.

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