This document provides examples and instructions for solving quadratic equations by completing the square. It begins with examples of solving quadratic equations using the square root property. It then explains how to complete the square to write a quadratic expression as a perfect square trinomial. Examples are provided to demonstrate completing the square and using it to solve quadratic equations. The document ensures readers understand completing the square through check examples and objectives.

1. 5-4 Completing the Square
5-4 Completing the Square
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra
Holt Algebra 2 2

2. 5-4
Completing the Square
Warm Up
Write each expression as a trinomial.
1. (x – 5)2
2. (3x + 5)2
x2 – 10x + 25
9x2 + 30x + 25
Factor each expression.
3. x2 – 18 + 81 (x – 9)2
4. 16x2 + 24x + 9 (4x + 3)2
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3. 5-4
Completing the Square
Objectives
Solve quadratic equations by completing the square.
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4. 5-4
Completing the Square
Vocabulary
completing the square
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5. 5-4
Completing the Square
Many quadratic equations contain expressions
that cannot be easily factored. For equations
containing these types of expressions, you can
use square roots to find roots.
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6. 5-4
Completing the Square
Reading Math
Read
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as “plus or minus square root of a.”

7. 5-4
Completing the Square
Example 1A: Solving Equations by Using the Square
Root Property
Solve the equation.
4x2 + 11 = 59
4x2 = 48
x2 = 12
Subtract 11 from both sides.
Divide both sides by 4 to isolate the
square term.
Take the square root of both sides.
Simplify.
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8. 5-4
Completing the Square
Example 1B: Solving Equations by Using the Square
Root Property
Solve the equation.
x2 + 12x + 36 = 28
(x + 6)2 = 28
Factor the perfect square trinomial
Take the square root of both sides.
Subtract 6 from both sides.
Simplify.
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9. 5-4
Completing the Square
Check It Out! Example 1a
Solve the equation.
4x2 – 20 = 5
4x2 = 25
x2 =
25
4
Add 20 to both sides.
Divide both sides by 4 to isolate the
square term.
Take the square root of both sides.
Simplify.
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10. 5-4
Completing the Square
Check It Out! Example 1b
Solve the equation.
x2 + 8x + 16 = 49
(x + 4)2 = 49
Factor the perfect square trinomial.
Take the square root of both sides.
x = –4 ± 49 Subtract 4 from both sides.
x = –11, 3
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Simplify.

11. 5-4
Completing the Square
If a quadratic expression of the form x2 + bx
cannot model a square, you can add a term to
form a perfect square trinomial. This is called
completing the square.
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12. 5-4
Completing the Square
Example 2A: Completing the Square
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 – 14x +
Find
.
Add.
x2 – 14x + 49
Factor.
(x – 7)2
Check Find the square of the binomial.
(x – 7)2 = (x – 7)(x – 7)
= x2 – 14x + 49
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13. 5-4
Completing the Square
Example 2B: Completing the Square
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 + 9x +
Find
Add.
Factor.
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.
Check Find the square
of the binomial.

14. 5-4
Completing the Square
Check It Out! Example 2a
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 + 4x +
Find
x2 + 4x + 4
(x + 2)2
Check
.
Add.
Factor.
Find the square of the binomial.
(x + 2)2 = (x + 2)(x + 2)
= x2 + 4x + 4
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15. 5-4
Completing the Square
Check It Out! Example 2b
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 – 4x +
Find
x2 – 4x + 4
(x – 2)2
Check
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.
Add.
Factor.
Find the square of the binomial.
(x – 2)2 = (x – 2)(x – 2)
= x2 – 4x + 4

16. 5-4
Completing the Square
Check It Out! Example 2c
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 + 3x +
Find
Add.
Factor.
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.
Check Find the square
of the binomial.

17. 5-4
Completing the Square
You can complete the square to solve quadratic
equations.
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18. 5-4
Completing the Square
Example 3A: Solving a Quadratic Equation by
Completing the Square
Solve the equation by completing the square.
x2 = 12x – 20
x – 12x = –20
Collect variable terms on
one side.
x2 – 12x +
Set up to complete the
square.
2
= –20 +
Add
x2 – 12x + 36 = –20 + 36
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to both sides.
Simplify.

19. 5-4
Completing the Square
Example 3A Continued
(x – 6)2 = 16
Factor.
Take the square root of
both sides.
x – 6 = ±4
x – 6 = 4 or x – 6 = –4
x = 10 or x = 2
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Simplify.
Solve for x.

20. 5-4
Completing the Square
Example 3B: Solving a Quadratic Equation by
Completing the Square
Solve the equation by completing the square.
18x + 3x2 = 45
x2 + 6x = 15
x2 + 6x +
= 15 +
Divide both sides by 3.
Set up to complete the
square.
Add
x2 + 6x + 9 = 15 + 9
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to both sides.
Simplify.

21. 5-4
Completing the Square
Example 3B Continued
(x + 3)2 = 24
Factor.
Take the square root of
both sides.
Simplify.
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22. 5-4
Completing the Square
Check It Out! Example 3a
Solve the equation by completing the square.
x2 – 2 = 9x
Collect variable terms on
one side.
x2 – 9x = 2
x2 – 9x +
=2+
Set up to complete the
square.
Add
to both sides.
Simplify.
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23. 5-4
Completing the Square
Check It Out! Example 3a Continued
Factor.
x–
9
= ± 89
4
2
x=
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9 ± 89
2
Take the square root of
both sides.
Simplify.

24. 5-4
Completing the Square
Check It Out! Example 3b
Solve the equation by completing the square.
3x2 – 24x = 27
Divide both sides by 3.
x2 – 8x = 9
x2 –8x +
=9+
Set up to complete the
square.
Add
to both sides.
Simplify.
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25. 5-4
Completing the Square
Check It Out! Example 3b Continued
Solve the equation by completing the square.
Factor.
Take the square root
of both sides.
Simplify.
x – 4 =–5 or x – 4 = 5
x =–1 or x = 9
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Solve for x.