# 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

Step #1:

Solve the quadratic equation to get x1 and x2

Step #2

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

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.

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