Name
LESSON
5-3
Date
Class
Reteach
Solving Quadratic Equations by Graphing and Factoring
Solve the equation ax 2 ⫹ bx ⫹ c ⫽ 0 to find the roots of the equation.
2
2
Find the roots of x ⫹ 2x ⫺ 15 ⫽ 0 to find the zeros of f x ⫽ x ⫹ 2x ⫺ 15.
x 2 ⫹ 2x ⫺ 15 ⫽ 0
Factor, then multiply
to check.
x ⫹ 5 x ⫺ 3 ⫽ 0
Solve each
equation for x.
Set each factor
equal to 0.
x ⫹ 5 ⫽ 0 or x ⫺ 3 ⫽ 0
x ⫽ ⫺5 or x ⫽ 3
To check the roots, substitute each root into the original equation:
Equation:
x 2 ⫹ 2x ⫺ 15 ⫽ 0
x 2 ⫹ 2x ⫺ 15 ⫽ 0
Root:
x ⫽ ⫺5
x⫽3
Check:
⫺5 2 ⫹ 2 ⫺5 ⫺ 15
3 2 ⫹ 2 3 ⫺ 15
25 ⫺ 10 ⫺ 15 ⫽ 0 ✓
9 ⫹ 6 ⫺ 15 ⫽ 0 ✓
The roots of the equation
are the zeros of the function.
2
The roots of x ⫹ 2x ⫺ 15 ⫽ 0 are ⫺5 and 3.
2
The zeros of f x ⫽ x ⫹ 2x ⫺ 15 are ⫺5 and 3.
Find the zeros of each function by factoring. Set the function equal to
0, factor, set each factor equal to 0, and then solve each equation.
2
1. f x ⫽ 4x ⫺ 24x
2. f x ⫽ x 2 ⫹ 4x ⫹ 3
4x 2 ⫺ 24x ⫽ 0
x 2 ⫹ 4x ⫹ 3 ⫽ 0
4x x ⫺
x ⫹ 3 x ⫹ 1 ⫽ 0
4x
6 ⫽0
⫽ 0 or x ⫺
6
⫽0
x⫹
x ⫽ 0 or x ⫽ 6
⫽ 0 or x ⫹
1
⫽0
x ⫽ ⫺3 or x ⫽ ⫺1
3. f x ⫽ x 2 ⫺ 5x ⫹ 4
4. f x ⫽ 3x 2 ⫹ 12x
2
3x ⫹ 12x ⫽ 0
2
⫺ 4 x ⫺ 1 ⫽ 0
3x x ⫹ 4 ⫽ 0
x ⫺ 5x ⫹ 4 ⫽ 0
x
3
x ⫺ 4 ⫽ 0 or x ⫺ 1 ⫽ 0
3x ⫽ 0 or x ⫹ 4 ⫽ 0
x ⫽ 4 or x ⫽ 1
x ⫽ 0 or x ⫽ ⫺4
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c05-3_rt.indd 22
22
Holt Algebra 2
12/20/05 3:14:52 PM
Process Black
Name
LESSON
5-3
Date
Class
Reteach
Solving Quadratic Equations by Graphing and Factoring
(continued)
Some quadratic equations have special factors.
2
2
Difference of Two Squares: a b a b a b Perfect Square Trinomials:
2
2
2
a 2ab b a b 2
2
2
a 2ab b a b Always write a quadratic equation in standard form before factoring.
16 and 25 are perfect squares.
Use the difference of two
squares to factor.
2
16x 25
2
16x 25 0
4x 2
5 2 0
4x 5 4x 5 0
4x 5 0 or 4x 5 0
Set each factor
equal to 0.
5
5 or x __
x __
4
4
Try to factor a perfect square trinomial if the coefficient of x and the
constant term are perfect squares.
4x 2 and 9 are
perfect squares.
2
4x 12x 9 0
2x 2
2 2x 3 3 2 0
2x 3 2x 3 2x 3 2 0
2x 3 0
The factors are
the same.
3
x __
2
Find the roots of each equation by factoring.
5. 4x 2 49
6. x 2 16 8x
0
4x 2 49 2
x 2
2x
x 8x 16 2
2
7 0
x 2 2 4 x 4 2 0
7 2x 7 0
2x 7 0 or 2x 7 0
7 or x __
7
x __
2
2
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c05-3_rt.indd 23
0
x
4 2 0
x
4 0
x4
23
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