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
Using the answers found in step #1, the factorization form is a (x  x
_{1})(x  x
_{2})
Two examples showing how to factor using the quadratic formula
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.

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