Solving Equations Using Factoring

Quadratic Equations

A quadratic equation is an equation that can be written in the standard form: ax² + bx + c = 0Quadratic equations will have zero, one, or two solutions.

Ways to Solve Quadratic Equations

Square Root Method3x2 = 108

Graphing

Quadratic Formula3x2 – 2x + 3 = 0

Factoring

Completing the Square

Solving by Factoring

x2 + 5x = -6Make it equal zerox2 + 5x + 6 = 0Factor the left side (use the box or one of the short cuts).(x + 2)(x + 3) = 0Set ALL FACTORS (including both sets of parentheses) equal to zero.

Solving by Factoring (cont.)

x + 2 = 0 x + 3 = 0Solve.x = -2 and x = -3

Example 2

x3 + x2 – 6x = 0Factor:x(x2 + x – 6) = 0x(x – 2)(x + 3)= 0Set each factor equal to zero:x = 0 x – 2 = 0 x + 3 = 0Solve:x = 0, 2, -3

Word Problem

You are building a rectangular wading pool. You want the area of the bottom to be 90 ft2. You want the length of the pool to be 3 ft longer than twice its width. What will be the dimensions of pool?

Word Problem (cont)

Draw a picture.w(2w + 3) = 90Distribute2w2 + 3w = 90Make it equal zero2w2 + 3w – 90 = 0Factor(2w + 15)(w – 6) = 0

w

2w + 3

Set each factor equal to zero2w + 15 = 0 w – 6 = 0Solve:W = -7.5 w = 6The width cannot be negative so it cannot be -7.5. It must be 6 feet.The length is 3 more than twice 6Dimensions are 6 feet by 15 feet

Try these…

x2 + 11x + 30 = 0

2x2 – 5x = 88

x2 – 5 = 4

3x2 + 4x = 2x2 – 2x – 9

x3 – 10x2 + 24x = 0

x = -5, -6x = -5.5, 8x = 3, -3x = -3x = 0, 6, 4

Try these (cont)…

You are building a rectangular wading pool. You want the area of the bottom to be 105 ft2. You want the length of the pool to be 1 ft longer than twice its width. What will be the dimensions of pool?7 feet by 15 feet