How do I know if a trinomial can be factored with integers?

Try finding two integers that multiply to c and add to b. If no such pair exists, the trinomial cannot be factored using integers. In that case, you will need the Quadratic Formula to solve equations. Not all trinomials factor nicely.

Why do both numbers have to be negative when b is negative and c is positive?

For the product pq to be positive, the two numbers must have the same sign (both positive or both negative). Since you also need p + q = b to be negative, both numbers must be negative. If they were both positive, their sum would be positive too.

Do I always get two different numbers p and q, or can they be the same?

They can be the same. For example, x^2 + 6x + 9 = (x+3)(x+3) = (x+3)^2. When p = q, the trinomial is a perfect square trinomial, and you have a repeated factor. This is a special case worth recognizing.

What is the difference between factoring and solving?

Factoring rewrites an expression as a product of factors, such as x^2 + 5x + 6 = (x+2)(x+3). Solving finds the values of x that make an equation true, such as solving x^2 + 5x + 6 = 0 to get x = -2 or x = -3. Factoring is a tool that helps you solve, but they are not the same thing.

Factoring Trinomials

Break a trinomial into two binomials by finding two numbers that add to b and multiply to c; use when solving equations or simplifying.

x^2 + bx + c = (x + p)(x + q), \quad p + q = b, ; pq = c

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What each symbol means

What Factoring Trinomials takes

Symbol	Meaning
$b$	The coefficient of $x$ in the original trinomial $x^2 + bx + c$ ; it must equal the sum $p + q$ of the two numbers you are looking for.
$c$	The constant term in the trinomial; it must equal the product $pq$ of the two numbers you are looking for, so if $c$ is negative, one of $p$ or $q$ must be negative.
$p$	One of the two numbers whose sum is $b$ and whose product is $c$ ; it appears in the binomial factor $(x + p)$ , and sign matters.
$q$	The other number whose sum is $b$ and whose product is $c$ ; it appears in the binomial factor $(x + q)$ , and switching the order does not change the result.
$x$	The variable (unknown value) in the trinomial; confusing it with its coefficient changes the entire meaning of the expression.

When to use it

When you need to solve a quadratic equation or simplify a polynomial expression.

Level

Usually taught in: Algebra I

Worked examples

1. Factor a simple trinomial

Problem

Factor

x^2 + 5x + 6

$x^2 + 5x + 6 \quad b = 5, \quad c = 6$
We identify the coefficients $b$ and $c$ in the form $x^2 + bx + c$ .
$2 + 3 = 5 \quad \text{and} \quad 2 \cdot 3 = 6$
We need two numbers that add to 5 and multiply to 6; the numbers 2 and 3 satisfy both conditions.
$x^2 + 5x + 6 = (x + 2)(x + 3)$
Each number becomes the constant term in a binomial factor with $x$ .

Answer: $(x + 2)(x + 3)$

Factoring is the reverse of expanding. We verify by expanding: $(x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$ .

2. Factor a trinomial with negative coefficients

Problem

Factor

x^2 - 7x + 12

$x^2 - 7x + 12 \quad b = -7, \quad c = 12$
Here $b = -7$ (negative), and $c = 12$ (positive).
$(-3) + (-4) = -7 \quad \text{and} \quad (-3) \cdot (-4) = 12$
For the product to be positive and the sum to be negative, both numbers must be negative.
$x^2 - 7x + 12 = (x - 3)(x - 4)$
When $p = -3$ and $q = -4$ , the binomials are $(x + p) = (x - 3)$ and $(x + q) = (x - 4)$ .

Answer: $(x - 3)(x - 4)$

When $b$ is negative and $c$ is positive, both factors must be negative. A negative sum and positive product only occur when both numbers have the same negative sign.

3. Find dimensions of a rectangular garden plot

Problem

A rectangular garden plot has an area of

x^2 + 9x + 20

square feet. By factoring, find the possible length and width dimensions.

$\text{Area} = x^2 + 9x + 20 \quad b = 9, \quad c = 20$
The area expression is in the form $x^2 + bx + c$ with $b = 9$ and $c = 20$ .
$4 + 5 = 9 \quad \text{and} \quad 4 \cdot 5 = 20$
Factor pairs of 20 are 1 and 20, 2 and 10, and 4 and 5; only 4 and 5 add to 9.
$x^2 + 9x + 20 = (x + 4)(x + 5)$
The length and width of the plot are represented by the two binomial factors.

Answer: $(x + 4)(x + 5)$

Factoring the area expression reveals the dimensions. When you multiply length times width, you get area, so if the area factors as $(x+4)(x+5)$ , the dimensions are $(x+4)$ feet and $(x+5)$ feet.

Common mistakes

Where Factoring Trinomials usually goes wrong

The mistake	Why it is wrong	The fix
For $x^2 + 5x + 6$ , writing $(x+2)(x+6)$ as the factorization.	Although 2 and 6 are factors of 6, they do not add to 5—only 2 and 3 do—so the sum condition is ignored.	Check both conditions: $2 + 3 = 5$ (sum) and $2 \cdot 3 = 6$ (product). Since $2 + 6 = 8 \neq 5$ , the factorization $(x+2)(x+6)$ is wrong.
For $x^2 - 7x + 12$ , finding the correct factors $(x-3)(x-4)$ but then writing $(x+3)(x+4)$ in the final answer.	The signs of the factors are lost or reversed after finding the numbers; positive factors are written instead of negative ones.	If the two numbers are -3 and -4, the binomials are $(x + (-3)) = (x - 3)$ and $(x + (-4)) = (x - 4)$ . Verify: $(x-3)(x-4) = x^2 - 7x + 12$ .
For $x^2 + 2x - 8$ , trying $(x+2)(x-4)$ as the factorization.	The numbers 2 and -4 multiply to -8 correctly, but they do not add to 2; instead $2 + (-4) = -2$ , giving the wrong coefficient for $x$ .	Verify both conditions. Although $2 \cdot (-4) = -8$ , the correct pair must satisfy $p + q = 2$ . The pair 4 and -2 works: $4 + (-2) = 2$ and $4 \cdot (-2) = -8$ , so the factorization is $(x+4)(x-2)$ .

Tips and when to use something else

Factoring trinomials works only when two integers exist that satisfy both conditions. For trinomials that do not factor with integers, use the Quadratic Formula instead.
Always verify your factorization by expanding $(x+p)(x+q)$ using the distributive property. If you get back the original trinomial, your answer is correct.
To find the factors faster, list all factor pairs of $c$ and calculate their sums. Only one pair will sum to $b$ ; do not guess.
The order of the factors does not matter because multiplication is commutative: $(x+2)(x+3) = (x+3)(x+2)$ .

Related formulas

Perfect Square TrinomialQuickly factor trinomials of the form a^2 \pm 2ab + b^2 by recognizing the perfect square pattern that gives (a \pm b)^2.Difference of SquaresInstantly factor binomials that are differences of perfect squares by recognizing the pattern and splitting into a product of a sum and difference.Quadratic FormulaThe Quadratic Formula solves any equation of the form ax^2 + bx + c = 0 in one step, giving both solutions at once without factoring.

Frequently asked questions

How do I know if a trinomial can be factored with integers?: Try finding two integers that multiply to $c$ and add to $b$ . If no such pair exists, the trinomial cannot be factored using integers. In that case, you will need the Quadratic Formula to solve equations. Not all trinomials factor nicely.
Why do both numbers have to be negative when $b$ is negative and $c$ is positive?: For the product $pq$ to be positive, the two numbers must have the same sign (both positive or both negative). Since you also need $p + q = b$ to be negative, both numbers must be negative. If they were both positive, their sum would be positive too.
Do I always get two different numbers $p$ and $q$ , or can they be the same?: They can be the same. For example, $x^2 + 6x + 9 = (x+3)(x+3) = (x+3)^2$ . When $p = q$ , the trinomial is a perfect square trinomial, and you have a repeated factor. This is a special case worth recognizing.
What is the difference between factoring and solving?: Factoring rewrites an expression as a product of factors, such as $x^2 + 5x + 6 = (x+2)(x+3)$ . Solving finds the values of $x$ that make an equation true, such as solving $x^2 + 5x + 6 = 0$ to get $x = -2$ or $x = -3$ . Factoring is a tool that helps you solve, but they are not the same thing.

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Reviewed 2026-09-18