Factoring Trinomials
Factoring Trinomials
Factoring trinomials is one of the most important skills in algebra. A trinomial of the form ax² + bx + c can often be written as the product of two binomials, and finding those binomials is the key to solving quadratic equations without the quadratic formula.
When the leading coefficient is 1, the technique is relatively simple — find two numbers that multiply to c and add to b. When the leading coefficient is not 1, the ac-method or trial-and-error approach is needed.
This lesson gives you systematic methods for both cases, along with plenty of practice to build speed and accuracy.
Simple Trinomials: $x^2 + bx + c$
Find two numbers that multiply to $c$ and add to $b$.
$x^2 + 7x + 12$
Product $= 12$, sum $= 7$: numbers $3$ and $4$. $(x + 3)(x + 4)$.
$x^2 - 5x - 14$
Product $= -14$, sum $= -5$: numbers $-7$ and $2$. $(x - 7)(x + 2)$.
$x^2 - 9x + 20$
Product $= 20$, sum $= -9$: numbers $-4$ and $-5$. $(x - 4)(x - 5)$.
General Trinomials: $ax^2 + bx + c$ ($a \neq 1$)
- Compute $a \cdot c$.
- Find two numbers with product $ac$ and sum $b$.
- Rewrite the middle term, then factor by grouping.
$6x^2 + 11x + 3$
- $ac = 18$. Numbers: $9$ and $2$ (product $18$, sum $11$).
- $6x^2 + 9x + 2x + 3$
- $3x(2x + 3) + 1(2x + 3)$
- $(2x + 3)(3x + 1)$
$2x^2 - 7x + 3$
$ac = 6$. Numbers: $-6$ and $-1$. $2x^2 - 6x - x + 3 = 2x(x-3) - 1(x-3) = (x-3)(2x-1)$.
Practice Problems
Show Answer Key
1. $(x+3)(x+5)$
2. $(x-5)(x+2)$
3. $(x+7)(x-5)$
4. $(x-6)^2$
5. $(2x+3)(x+1)$
6. $(3x-4)(x-2)$
7. $(x+7)(x-6)$
8. $(2x+3)(2x-1)$
9. $(x-2)(x-9)$
10. $(5x-2)(x+3)$
11. $(2x+1)(3x-2)$
12. $(x+3)^2$