Factoring

When a is not 1

Example 1

Factor and solve 2x2 + x - 6 = 0.

Step 1. Multiply a and c.

In this equation, a = 2 and c = -6, therefore ac = -12.

Step 2. List all factor pairs of ac.

In this equation ac = -12.
The factor pairs of -12 are 1 and -12, -1 and 12, 2 and -6, -2 and 6, 3 and -4, -3 and 4.

Step 3. Determine which factor pair has the sum of b.

In this equation b = 1.
The sum of the factor pairs are 1 + -12 = -11, -1 + 12 = 11, 2 + -6 = -4, -2 + 6 = 4, 3 + -4 = -1, -3 + 4 = 1.

4 and -3 have the correct sum.

Step 4. Substitute the roots into the original equation in place of b.

2x2 + (4 + -3)x - 6 = 0

2x2 + 4x + -3x - 6 = 0

Step 5. Factor by grouping the first two terms and the last two terms together and find their GCF.

(2x2 + 4x) + (-3x - 6) = 0

2x(x + 2) -3(x + 2) = 0

(x + 2)(2x - 3) = 0

Step 6. Use the Zero Product Property to set each factor equal to zero and solve.

x + 2 = 0 or 2x - 3 = 0

x = -2 or x = 1.5