# Solving Quadratic Equations by Factoring

Example 1

Solve by factoring: x2 - 14 = -5x

Solution

 Step 1 Write the equation in the form ax2 + bx + c = 0Add 5x to both sides. Step 2 Factor the polynomial. The coefficient of x2 is 1, so find two numbers whose product is -14 and whose sum is 5. x2 + 5x - 14 = 0 These numbers are -2 and 7. Step 3 Use the Zero Product Property. Step 4 Solve the resulting equations. Step 5 Check each answer. We leave the check to you. (x - 2)(x + 7)x - 2 = 0 x = 2 = 0or x + 7 = 0 or x = -7
So, the two solutions of x2 - 14 = -5x are x = 2 and x = -7.

Example 2

Solve by factoring: 7x + 6x2 = 20

Solution

 Step 1 Write the equation in the form ax2 + bx + c = 0Subtact 20 from both sides. Step 2 Factor the polynomial. The coefficient of x2 is not 1, so find two numbers whose product is ac and whose sum is b. ac = 6(-20) = -120 , b = 7 The two numbers are -8 and 15. 6x2 + 7x - 20 = 0 Rewrite the middle term, 7x, as -8x + 15x. Factor 2x from the first pair of terms. Factor 5 from the second pair of terms. Factor out the common binomial factor, 3x - 4.Step 3 Use the Zero Product Property. Step 4 Solve the resulting equations. Step 5 Check each answer. We leave the check to you. 6x2 - 8x + 15x - 20   2x(3x - 4) + 5(3x - 4) (3x - 4)(2x + 5) 3x - 4 = 0 or 2x + 5 = 0  = 0 = 0= 0
So, the two solutions of 7x + 6x2 = 20 are