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.
Slide 5.5-3
Objective 1
Solve quadratic equations by factoring.
Slide 5.5-4
Solve quadratic equations by factoring.
We use the zero-factor property to solve a quadratic equation by factoring.
Slide 5.5-5
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
Slide 5.5-6
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
Slide 5.5-7
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.
Slide 5.5-8
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.
Slide 5.5-9
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
Slide 5.5-10
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
Slide 5.5-11
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
Slide 5.5-12
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.
Slide 5.5-13
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
Slide 5.5-14
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:
Slide 5.5-15
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:
Slide 5.5-16
Solving an Equation Requiring Multiplication before FactoringCLASSROOM EXAMPLE 7