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 x
_{1} and x
_{2}
Step #2
Uisng the answers found in step #1, the factorization form is a (x - x
_{1})(x - x
_{2})
Example #1:
Factor 4x
^{2} + 9x + 2 = 0 using the quadratic formula.
a = 4, b = 9, and c = 2
x = (-b ± √(b
^{2} - 4ac)) / 2a
x = (-9 ± √(9
^{2} - 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
x
_{1} = (-9 + 7 ) / 8
x
_{1} = (-2 ) / 8
x
_{1} = -1/4
x
_{2} = (-9 - 7 ) / 8
x
_{2} = (-16 ) / 8
x
_{2} = -2
The factorization form is a (x - x
_{1})(x - x
_{2})
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 x
^{2} + 2x - 15 = 0 using the quadratic formula
a = 1, b = 2, and c = -15
x = (-b ± √(b
^{2} - 4ac)) / 2a
x = (- 2 ± √(2
^{2} - 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
x
_{1} = (-2 + 8 ) / 2
x
_{1} = ( 6 ) / 2
x
_{1} = 3
x
_{2} = (-2 - 8 ) / 2
x
_{2} = (-10) / 2
x
_{2} = -5
The factorization form is a (x - x
_{1})(x - x
_{2})
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.