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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 = 0

Add 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

= 0

or 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 = 0

Subtact 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
 
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