5.5 Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Factoring.

Quadratic EquationA quadratic equation is an equation that can be written in the form

ax2 + bx + c = 0, where a, b, and c are real numbers, with a ≠ 0.

The form ax2 + bx + c = 0 is the standard form of a quadratic equation. For example,

and

are all quadratic equations, but only x2 + 5x +6 = 0 is in standard form.

2 5 6 0,x x 22 5 3,x x 2 4x

Until now, we have factored expressions, including many quadratic expressions. In this section we see how we can use factored quadratic expressions to solve quadratic equations.

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Objective 1

Solve quadratic equations by factoring.

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Solve quadratic equations by factoring.

We use the zero-factor property to solve a quadratic equation by factoring.

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Zero-Factor PropertyIf a and b are real numbers and if ab = 0, then a = 0 or b = 0.

That is, if the product of two numbers is 0, then at least one of the numbers must be 0. One number must, but both may be 0.

Solve.

Solution:

2 3 5 7 0x x

2 4 0x x

2 3 0x

0x

32 33 0x 2 3

2 2

x

3

2x

75 77 0x 5 7

5 5

x

7

5x

5 7 0x

2 4 0x 42 44 0x

2 4

2 2

x

2x

0, 2

3 7,

2 5

or

or

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Using the Zero-Factor PropertyCLASSROOM EXAMPLE 1

Solve each equation.Solution:

2 2 8x x

2 30x x

4 0x

6 0x

4 44 0x 4x

2 22 0x 2x

2 0x

5 0x 6 66 0x

6x 5 55 0x

5,6

2 8 82 8x x 2 2 8 0x x

4 2 0x x 4,2

2 3 30030x xx x 2 30 0x x

6 5 0x x 5x

or

or

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Solving Quadratic EquationsCLASSROOM EXAMPLE 2

Solving a Quadratic Equation by Factoring

Step 1: Write the equation in standard form — that is, with all terms on one side of the equals sign in descending power of the variable and 0 on the other side.

Step 2: Factor completely.

Step 3: Use the zero-factor property to set each factor with variable equal to 0, and solve the resulting equations.

Step 4: Check each solution in the original equation.

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Solve quadratic equations by factoring. (cont’d)

Solution:

Solve 3m2 − 9m = 30.

2 303 9 3 300m m

2 5 0m m

23 9 30 0m m

2 3 10 03 m m

2 0m 2 22 0m

2m

5 0m

5 55 0m

5m 2,5

A common error is to include the common factor 3 as a solution. Only factors containing variables lead to solutions.

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Solving a Quadratic Equation with a Common FactorCLASSROOM EXAMPLE 3

2 3 10 0m m

Solution:

249 9 0x

7 3 7 3 0x x

2 3x x

3 3,

7 7

3 33 0x

Solve each equation.

0,3

2 333x xx x 3 0x x

0x

3x

37 33 0x

3

7x

7 3

7 7

x

37 33 0x 7 3

7 7

x

3

7x

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Solving Quadratic EquationsCLASSROOM EXAMPLE 4

Solution:

Solve the equation.

4 7 2x x

24 2 27 2x x 24 7 2 0x x

2 4 1 0x x

12,

4

2 22 0x 2x

2 0x

4 1

4 4

x

14 11 0x

1

4x

4 1 0x

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Solving Quadratic Equations (cont’d)CLASSROOM EXAMPLE 4

Solution:

2 16 64x x

8 8 0x x

Solve.

88 0 or 8 0x x

8x

2 16 64 0x x

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Solving Quadratic Equations with Double Solutions

There is no need to write the same number more than once in a solution set when a double solution occurs.

CLASSROOM EXAMPLE 5

Objective 2

Solve other equations by factoring.

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Solve the equation.

Solution:

32 50 0x x

22 25 0x x

2 5 5 0x x x

2 0x 2 0

2 2

x

0x 0, 5,5

5 55 0x 5x

5 55 0x 5x

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Solving Equations with More than Two Variable FactorsCLASSROOM EXAMPLE 6

Solve the equation.

22 1 2 7 15 0x x x

2 1 2 3 5 0x x x

12 11 0x 2 1

2 2

x

1

2x

32 33 0x 2 3

2 2

x

3

2x

5 55 0x

1 35, ,

2 2

5x

Solution:

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Solving Equations with More Than Two Variable Factors (cont’d)CLASSROOM EXAMPLE 6

Solve.

21 2 1 1x x x

2 22 3 1 2 1x x x x 2 22 22 1 2 12 3 1 2 1x x x xx x x x

2 5 0x x 5 0x x

0x 5 55 0x 5x 0,5

Solution:

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Solving an Equation Requiring Multiplication before FactoringCLASSROOM EXAMPLE 7