Factoring Quadratics

Definition: A quadratic equation is an equation in the form ax² + bx + c = 0. If you need a refresher on quadratic equations, click here.

Factoring a quadratic (leading coefficient of 1):
To factor a quadratic in the form x² + bx + c, find two factors of c whose sum is b.

Factoring a quadratic (leading coefficient ≠ 1):
To factor a quadratic in the form ax² + bx + c (where a ≠ 1), after pulling out any common factors, find two numbers whose product is equal to the product if a and c, and whose sum is equal to b.

Example: Factor the following quadratic: x² + 19x + 60

Solution: 

Step 1: Notice that this quadratic has a leading coefficient of 1, so we will find two numbers whose product is c (60), and whose sum is b (19).

Listing the factors of 60 will allow us to choose the best pair of numbers:

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Since we are looking for a pair of numbers whose sum is 19, 15 and 4 will work here.

Step 2: Using the chosen number of pair, write the expression in the form (x + _) (x + _) using these numbers in the blanks:

(x +15)(x + 4)

Step 3: Use the FOIL method to check this answer:

(x + 15)(x + 4) = x*x + 15*x + 4*x + 15*4 = x² + 15x + 4x + 60 = x² + 19x + 60

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