Completing the Square

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9-8
Completing the Square
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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• Warm Up
• Simplify.

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1.
2.
3.
4.
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• Warm Up
• Solve each quadratic equation by factoring.
• 5. x2 8x 16 0
• 6. x2 22x 121 0
• 7. x2 12x 36 0

x 4
x 11
x 6
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Objective
Solve quadratic equations by completing the
square.
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Vocabulary
completing the square
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In the previous lesson, you solved quadratic
equations by isolating x2 and then using square
roots. This method works if the quadratic
equation, when written in standard form, is a
perfect square.
When a trinomial is a perfect square, there is a
relationship between the coefficient of the
x-term and the constant term.
X2 6x 9 x2 8x 16
Divide the coefficient of the x-term by 2, then
square the result to get the constant term.
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An expression in the form x2 bx is not a
perfect square. However, you can use the
relationship shown above to add a term to x2 bx
to form a trinomial that is a perfect square.
This is called completing the square.
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Example 1 Completing the Square
Complete the square to form a perfect square
trinomial.
x2 2x
x2 6x
Identify b.
x2 2x 1
x2 6x 9
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Check It Out! Example 1
Complete the square to form a perfect square
trinomial.
x2 12x
x2 5x
Identify b.
x2 12x 36
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Check It Out! Example 1
Complete the square to form a perfect square
trinomial.
x2 8x
Identify b.
x2 12x 16
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To solve a quadratic equation in the form x2 bx
c, first complete the square of x2 bx. Then
you can solve using square roots.
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Solving a Quadratic Equation by Completing the
Square
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Example 2A Solving x2 bx c
Solve by completing the square.
x2 16x 15
The equation is in the form x2 bx c.
Step 1 x2 16x 15
Step 3 x2 16x 64 15 64
Complete the square.
Step 4 (x 8)2 49
Factor and simplify.
Take the square root of both sides.
Step 5 x 8 7
Write and solve two equations.
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Example 2A Continued
Solve by completing the square.
x2 16x 15
The solutions are 1 and 15.
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Example 2B Solving x2 bx c
Solve by completing the square.
x2 4x 6 0
Write in the form x2 bx c.
Step 1 x2 (4x) 6
Step 3 x2 4x 4 6 4
Complete the square.
Step 4 (x 2)2 10
Factor and simplify.
Take the square root of both sides.
Write and solve two equations.
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Example 2B Continued
Solve by completing the square.
The solutions are 2 v10 and x 2 v10.
Check Use a graphing calculator to check your
answer.
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Check It Out! Example 2a
Solve by completing the square.
x2 10x 9
The equation is in the form x2 bx c.
Step 1 x2 10x 9
Step 3 x2 10x 25 9 25
Complete the square.
Factor and simplify.
Step 4 (x 5)2 16
Take the square root of both sides.
Step 5 x 5 4
Write and solve two equations.
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Check It Out! Example 2a Continued
Solve by completing the square.
x2 10x 9
The solutions are 9 and 1.
Check
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Check It Out! Example 2b
Solve by completing the square.
t2 8t 5 0
Write in the form x2 bx c.
Step 1 t2 (8t) 5
Step 3 t2 8t 16 5 16
Complete the square.
Factor and simplify.
Step 4 (t 4)2 21
Take the square root of both sides.
Write and solve two equations.
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Check It Out! Example 2b Continued
Solve by completing the square.
The solutions are
t 4 v21 or t 4 v21.
Check Use a graphing calculator to check your
answer.
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Example 3A Solving ax2 bx c by Completing
the Square
Solve by completing the square.
3x2 12x 15 0
Divide by 3 to make a 1.
Write in the form x2 bx c.
x2 (4x) 5
Complete the square.
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Example 3A Continued
Solve by completing the square.
3x2 12x 15 0
Factor and simplify.
There is no real number whose square is negative,
so there are no real solutions.
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Example 3B Solving ax2 bx c by Completing
the Square
Solve by completing the square.
5x2 19x 4
Step 1
Divide by 5 to make a 1.
Write in the form x2 bx c.
Step 2
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Example 3B Continued
Solve by completing the square.
Complete the square.
Step 3
Rewrite using like denominators.
Factor and simplify.
Take the square root of both sides.
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Example 3B Continued
Solve by completing the square.
Write and solve two equations.
Step 6
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Check It Out! Example 3a
Solve by completing the square.
3x2 5x 2 0
Step 1
Divide by 3 to make a 1.
Write in the form x2 bx c.
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Check It Out! Example 3a Continued
Solve by completing the square.
Step 2
Complete the square.
Factor and simplify.
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Check It Out! Example 3a Continued
Solve by completing the square.
Step 5
Take the square root of both sides.
Write and solve two equations.
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Check It Out! Example 3b
Solve by completing the square.
4t2 4t 9 0
Step 1
Divide by 4 to make a 1.
Write in the form x2 bx c.
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Check It Out! Example 3b Continued
Solve by completing the square.
4t2 4t 9 0
Step 2
Complete the square.
Step 4
Factor and simplify.
There is no real number whose square is negative,
so there are no real solutions.
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Example 4 Problem-Solving Application
A rectangular room has an area of 195 square
feet. Its width is 2 feet shorter than its
length. Find the dimensions of the room. Round to
the nearest hundredth of a foot, if necessary.
The answer will be the length and width of the
room.
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Example 4 Continued
Set the formula for the area of a rectangle equal
to 195, the area of the room. Solve the equation.
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Example 4 Continued
Let x be the width. Then x 2 is the length.
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Example 4 Continued
Step 1 x2 2x 195
Simplify.
Complete the square by adding 1 to both sides.
Step 3 x2 2x 1 195 1
Factor the perfect-square trinomial.
Step 4 (x 1)2 196
Take the square root of both sides.
Step 5 x 1 14
Step 6 x 1 14 or x 1 14
Write and solve two equations.
x 13 or x 15
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Example 4 Continued
Negative numbers are not reasonable for length,
so x 13 is the only solution that makes sense.
The width is 13 feet, and the length is 13 2,
or 15, feet.
Look Back
The length of the room is 2 feet greater than the
width. Also 13(15) 195.
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Check It Out! Example 4
An architect designs a rectangular room with an
area of 400 ft2. The length is to be 8 ft longer
than the width. Find the dimensions of the room.
Round your answers to the nearest tenth of a
foot.
The answer will be the length and width of the
room.
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Check It Out! Example 4 Continued
Set the formula for the area of a rectangle equal
to 400, the area of the room. Solve the equation.
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Check It Out! Example 4 Continued
Let x be the width. Then x 8 is the length.
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Check It Out! Example 4 Continued
Step 1 x2 8x 400
Simplify.
Step 3 x2 8x 16 400 16
Complete the square by adding 16 to both sides.
Step 4 (x 4)2 416
Factor the perfect-square trinomial.
Step 5 x 4 ? 20.4
Take the square root of both sides.
Step 6 x 4 ? 20.4 or x 4 ? 20.4
Write and solve two equations.
x ? 16.4 or x ? 24.4
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Check It Out! Example 4 Continued
Negative numbers are not reasonable for length,
so x ? 16.4 is the only solution that makes sense.
The width is approximately16.4 feet, and the
length is 16.4 8, or approximately 24.4, feet.
Look Back
The length of the room is 8 feet longer than the
width. Also 16.4(24.4) 400.16, which is
approximately 400.
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Lesson Quiz Part I
Complete the square to form a perfect square
trinomial. 1. x2 11x 2. x2 18x Solve
by completing the square. 3. x2 2x 1 0 4.
3x2 6x 144 5. 4x2 44x 23
81
6, 8
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Lesson Quiz Part II
6. Dymond is painting a rectangular banner for a
football game. She has enough paint to cover 120
ft2. She wants the length of the banner to be 7
ft longer than the width. What dimensions should
Dymond use for the banner?
8 feet by 15 feet