Factoring Quadratics

Factoring is multiplying, reversed

Multiplying turns a product into a sum:

$$(x + 3)(x + 5) = x^2 + 8x + 15$$

Factoring runs the same line right to left. You are handed $x^2 + 8x + 15$ and asked which two sets of parentheses produced it.

That is why the multiplying comes first. Every factoring method is a way of reading the box backwards.

Why a pair of numbers is what you are looking for

Look at where the $8$ and the $15$ came from.

Comes from	Product
$15$	$3 \times 5$
$8x$	$3x + 5x$

So the constant is the two numbers multiplied, and the middle coefficient is the two numbers added.

$$x^2 + bx + c = (x + p)(x + q) \quad\text{where}\quad pq = c \;\text{ and }\; p + q = b$$

To factor, find that pair.

Factor $x^2 + 7x + 12$.

List the factor pairs of $12$ and check which adds to $7$:

Pair	Product	Sum
$1, 12$	$12$	$13$
$2, 6$	$12$	$8$
$3, 4$	$12$	$7$

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

Start from the multiplication. A number has only a handful of factor pairs, while endless pairs add to $7$. Working the short list first is what makes this quick rather than a search.

Getting the signs right

The signs of $c$ and $b$ decide the signs of the pair before any hunting starts.

$c$	$b$	The pair
positive	positive	both positive
positive	negative	both negative
negative	either	one of each sign

A positive $c$ means the two numbers have the same sign, since only matching signs multiply to a positive. Which sign is then settled by $b$.

A negative $c$ means opposite signs, and the larger number carries the sign of $b$.

Factor $x^2 - 2x - 15$.

$c$ is negative, so one of each sign. Pairs multiplying to $-15$: $(1, -15)$, $(-1, 15)$, $(3, -5)$, $(-3, 5)$. The one adding to $-2$ is $3$ and $-5$.

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

Take out a common factor first

Before anything else, check whether all three terms share a factor.

$$2x^2 + 10x + 12 = 2(x^2 + 5x + 6) = 2(x + 2)(x + 3)$$

Pulling the $2$ out leaves a leading coefficient of $1$, which is the case the pair method handles directly. Skipping this step leaves you hunting through the factor pairs of $12$ with a $2$ in the way.

Common factor, then pattern, then pair. In that order, each step makes the next one smaller.

The two patterns to spot on sight

Difference of squares — two square terms with a minus between them, and no middle term:

$$a^2 - b^2 = (a - b)(a + b)$$ $$x^2 - 36 = (x - 6)(x + 6)$$

There is no pair to hunt for. The absence of a middle term is the signal, and it happens because $+6x$ and $-6x$ cancel.

Note that $x^2 + 36$ does not factor this way. A sum of squares has no factors with real numbers in them.

Perfect square trinomial — first and last terms both squares, middle term twice the product of their roots:

$$a^2 + 2ab + b^2 = (a + b)^2$$ $$x^2 + 10x + 25 = (x + 5)^2$$

Check the middle: $2 \times 5 = 10$. It matches, so the square form is right.

These two are worth recognizing because they arrive constantly, and each saves the whole search.

Checking the answer

Multiply the parentheses back out. The result has to be the original quadratic, every term.

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

This check is complete — it cannot pass a wrong factoring — and it takes one line. There is no reason to skip it.

Worked examples

Common mistakes

Practice problems