Factoring using the quadratic formula

Factoring using the quadratic formula is the goal of this lesson. It is closely related to solving equations using the quadratic formula.

2 easy steps to follow when factoring using the quadratic formula:

Step #1:

Solve the quadratic equation to get x₁ and x₂

Step #2

Using the answers found in step #1, the factorization form is a (x - x₁)(x - x₂)

Two examples showing how to factor using the quadratic formula

Example #1:


Factor 4x² + 9x + 2 = 0 using the quadratic formula.

a = 4, b = 9, and c = 2

x = (-b ± √(b² - 4ac)) / 2a

x = (-9 ± √(9² - 4 × 4 × 2)) / 2 × 4

x = (-9 ± √(81 - 4 × 4 × 2)) / 8

x = (-9 ± √(81 - 4 × 8)) / 8

x = (-9 ± √(81 - 32)) / 8

x = (-9 ± √(49)) / 8

x = (-9 ± 7 ) / 8

x₁ = (-9 + 7 ) / 8

x₁ = (-2 ) / 8

x₁ = -1/4

x₂ = (-9 - 7 ) / 8

x₂ = (-16 ) / 8

x₂ = -2


The factorization form is a (x - x₁)(x - x₂)

The factorization form is 4 (x - -1/4)(x - -2)

The factorization form is 4 (x + 1/4)(x + 2)

Now, use distributive property to simplify the expression by getting rid of fractions

4 (x + 1/4)(x + 2) = (4 × x + 4 × 1/4) (x + 2) = (4x + 1)(x + 2)


Example #2:


Factor x² + 2x - 15 = 0 using the quadratic formula

a = 1, b = 2, and c = -15

x = (-b ± √(b² - 4ac)) / 2a

x = (- 2 ± √(2²  - 4 × 1 × -15)) / 2 × 1

x = (-2 ± √(4 - 4 × 1 × -15)) / 2

x = (-2 ± √(4 - 4 × -15)) / 2

x = (-2 ± √(4 + 60)) / 2

x = (-2 ± √(64)) / 2

x = (-2 ± 8 ) / 2

x₁ = (-2 + 8 ) / 2

x₁ = ( 6 ) / 2

x₁ = 3

x₂ = (-2 - 8 ) / 2

x₂ = (-10) / 2

x₂ = -5

The factorization form is a (x - x₁)(x - x₂)

The factorization form is 1 (x - 3)(x - -5)

The factorization form is 1 (x - 3)(x + 5)

Now, use distributive property to simplify the expression

1 (x - 3)(x + 5) = (1 × x + 1 × -3) (x + 2) = (x - 3)(x + 5)

It is important to understand how to use the quadratic formula before fatoring using the quadratic formula.