What if the quadratic cannot be factored with integers?

Not every quadratic factors neatly over the integers. Check the discriminant $b^2 - 4ac$: if it is not a perfect square, the expression is called prime (unfactorable over the integers), and you should use the quadratic formula or completing the square instead.

Factoring Quadratics — Definition, Formula & Examples

Factoring quadratics is the process of rewriting a quadratic expression like

ax^2 + bx + c

as a product of two simpler expressions (binomials). For example,

x^2 + 5x + 6

factors into

(x + 2)(x + 3)

To factor a quadratic polynomial $ax^2 + bx + c$ over the integers is to express it as $a(x - r)(x - s)$ , where $r$ and $s$ are the roots of the polynomial satisfying $r + s = -\tfrac{b}{a}$ and $r \cdot s = \tfrac{c}{a}$ . When $a = 1$ , this reduces to finding integers $r$ and $s$ such that $r + s = -b$ and $rs = c$ .

Key Formula

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

Where:

$b$ = Coefficient of the linear term
$c$ = Constant term
$p, q$ = Two numbers whose sum is b and whose product is c

How It Works

When

a = 1

, look for two numbers that multiply to

c

and add to

b

. Call them

p

and

q

; then

x^2 + bx + c = (x + p)(x + q)

. When

a \neq 1

, one reliable method is the AC method: multiply

a \cdot c

, find two numbers that multiply to

ac

and add to

b

, then split the middle term and factor by grouping. You can always verify your answer by multiplying the binomials back out using the FOIL method.

Worked Example

Problem: Factor

x^2 + 7x + 12

Identify b and c: Here

b = 7

and

c = 12

. You need two numbers that add to 7 and multiply to 12.

p + q = 7, \quad p \cdot q = 12

Find the pair: List factor pairs of 12:

(1,12),\,(2,6),\,(3,4)

. The pair

3

and

4

adds to

7

3 + 4 = 7 \quad \checkmark

Write the factored form: Place the two numbers into the binomials.

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

Verify by expanding: Use FOIL to confirm:

x^2 + 4x + 3x + 12 = x^2 + 7x + 12

\checkmark

Answer:

(x + 3)(x + 4)

Another Example

Problem: Factor

2x^2 + 7x + 3

using the AC method.

Compute a · c: Multiply the leading coefficient by the constant:

2 \times 3 = 6

. Find two numbers that multiply to 6 and add to 7.

p \cdot q = 6, \quad p + q = 7

Find the pair: The numbers 1 and 6 work:

1 \times 6 = 6

and

1 + 6 = 7

Split the middle term: Rewrite

7x

x + 6x

and group.

2x^2 + x + 6x + 3 = x(2x + 1) + 3(2x + 1)

Factor out the common binomial: Both groups share the factor

(2x + 1)

= (2x + 1)(x + 3)

Answer:

(2x + 1)(x + 3)

Why It Matters

Factoring quadratics is a core skill in Algebra 1 and Algebra 2, appearing on virtually every standardized test including the SAT and ACT. Engineers and scientists use it to find where a parabolic model equals zero—for instance, determining when a projectile hits the ground. Mastering factoring also builds the foundation for working with polynomial division and rational expressions in precalculus.

Common Mistakes

Mistake: Forgetting to account for signs when

c

is negative

Correction: If

c < 0

, the two numbers

p

and

q

have opposite signs. Focus on which combination gives the correct sign for

b

. For example, in

x^2 + 2x - 15

, the pair is

5

and

-3

(not

-5

and

3

) because

5 + (-3) = +2

Mistake: Not factoring out the GCF first when

a \neq 1

Correction: Always check for a greatest common factor before applying the AC method. For

3x^2 + 12x + 12

, first factor out

3

to get

3(x^2 + 4x + 4)

, then factor the simpler trinomial as

3(x+2)^2

Related Terms

Factored Form — The result you get after factoring
FOIL Method — Used to verify factoring by expanding
Quadratic Equation — Set factored form to zero to solve
Completing the Square — Alternative method when factoring fails
Discriminant of a Quadratic — Tells whether integer factoring is possible
Perfect Square Trinomial — Special case that factors as a squared binomial
Double Root — Occurs when both factors are identical
Parabola — Graph whose x-intercepts come from factoring