Download completing the square

9-8
9-8 Completing
Completingthe
theSquare
Square
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
11
9-8 Completing the Square
Warm Up
Simplify.
1.
2.
3.
4.
Holt Algebra 1
19
9-8 Completing the Square
Warm Up
Solve each quadratic equation by
factoring.
5. x2 + 8x + 16 = 0 x = –4
6. x2 – 22x + 121 = 0 x = 11
7. x2 – 12x + 36 = 0
Holt Algebra 1
x=6
9-8 Completing the Square
Objective
Solve quadratic equations by
completing the square.
Holt Algebra 1
9-8 Completing the Square
Vocabulary
completing the square
Holt Algebra 1
9-8 Completing the Square
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
Holt Algebra 1
x2 – 8x + 16
Divide the coefficient of
the x-term by 2, then
square the result to get
the constant term.
9-8 Completing the Square
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.
Holt Algebra 1
9-8 Completing the Square
Example 1: Completing the Square
Complete the square to form a perfect square
trinomial.
A. x2 + 2x +
x2 + 2x
B. x2 – 6x +
Identify b.
x2 + –6x
.
x2 + 2x + 1
Holt Algebra 1
x2 – 6x + 9
9-8 Completing the Square
Check It Out! Example 1
Complete the square to form a perfect square
trinomial.
a. x2 + 12x +
x2 + 12x
b. x2 – 5x +
Identify b.
x2 + –5x
.
x2 + 12x + 36
Holt Algebra 1
x2 – 6x +
9-8 Completing the Square
Check It Out! Example 1
Complete the square to form a perfect square
trinomial.
c. 8x + x2 +
x2 + 8x
Identify b.
.
x2 + 12x + 16
Holt Algebra 1
9-8 Completing the Square
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.
Holt Algebra 1
9-8 Completing the Square
Solving a Quadratic Equation by Completing the Square
Holt Algebra 1
9-8 Completing the Square
Example 2A: Solving x2 +bx = c
Solve by completing the square.
x2 + 16x = –15
Step 1 x2 + 16x = –15
Step 2
The equation is in the
form x2 + bx = c.
.
Step 3 x2 + 16x + 64 = –15 + 64
Complete the square.
Step 4 (x + 8)2 = 49
Factor and simplify.
Step 5 x + 8 = ± 7
Take the square root
of both sides.
Write and solve two
equations.
Step 6 x + 8 = 7 or x + 8 = –7
x = –1 or
x = –15
Holt Algebra 1
9-8 Completing the Square
Example 2A Continued
Solve by completing the square.
x2 + 16x = –15
The solutions are –1 and –15.
Check x2 + 16x = –15
(–1)2 + 16(–1)
1 – 16
–15
Holt Algebra 1
–15
–15
–15
x2 + 16x = –15
(–15)2 + 16(–15)
225 – 240
–15
–15
–15
–15
9-8 Completing the Square
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 2
.
Step 3 x2 – 4x + 4 = 6 + 4
Complete the square.
Step 4 (x – 2)2 = 10
Factor and simplify.
Step 5 x – 2 = ± √10
Take the square root
of both sides.
Step 6 x – 2 = √10 or x – 2 = –√10 Write and solve two
x = 2 + √10 or x = 2 – √10 equations.
Holt Algebra 1
9-8 Completing the Square
Example 2B Continued
Solve by completing the square.
The solutions are 2 + √10 and x = 2 – √10.
Check Use a
graphing calculator
to check your
answer.
Holt Algebra 1
9-8 Completing the Square
Check It Out! Example 2a
Solve by completing the square.
x2 + 10x = –9
Step 1 x2 + 10x = –9
Step 2
Step 3 x2 + 10x + 25 = –9 + 25
Step 4 (x + 5)2 = 16
Step 5 x + 5 = ± 4
Step 6 x + 5 = 4 or x + 5 = –4
x = –1 or
x = –9
Holt Algebra 1
The equation is in the
form x2 + bx = c.
.
Complete the square.
Factor and simplify.
Take the square root
of both sides.
Write and solve two
equations.
9-8 Completing the Square
Check It Out! Example 2a Continued
Solve by completing the square.
x2 + 10x = –9
The solutions are –9 and –1.
Check
x2 + 16x = –15
(–1)2 + 16(–1)
1 – 16
–15
Holt Algebra 1
–15
–15
–15
x2 + 10x = –9
(–9)2 + 10(–9)
–9
81 – 90
–9
–9
–9

9-8 Completing the Square
Check It Out! Example 2b
Solve by completing the square.
t2 – 8t – 5 = 0
Step 1 t2 + (–8t) = 5
Step 2
Write in the form
x2 + bx = c.
.
Step 3 t2 – 8t + 16 = 5 + 16
Complete the square.
Step 4 (t – 4)2 = 21
Factor and simplify.
Step 5 t – 4 = ± √21
Take the square root
of both sides.
Step 6 t = 4 + √21 or t = 4 – √21
Write and solve two
equations.
Holt Algebra 1
9-8 Completing the Square
Check It Out! Example 2b Continued
Solve by completing the square.
The solutions are t = 4 – √21 or t = 4 + √21.
Check Use a
graphing calculator
to check your
answer.
Holt Algebra 1
9-8 Completing the Square
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.
Step 1
x2 – 4x + 5 = 0
x2 – 4x = –5
x2 + (–4x) = –5
Step 2
Write in the form x2 + bx = c.
.
Step 3 x2 – 4x + 4 = –5 + 4 Complete the square.
Holt Algebra 1
9-8 Completing the Square
Example 3A Continued
Solve by completing the square.
–3x2 + 12x – 15 = 0
Step 4 (x – 2)2 = –1
Factor and simplify.
There is no real number whose square is
negative, so there are no real solutions.
Holt Algebra 1
9-8 Completing the Square
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
Holt Algebra 1
.
9-8 Completing the Square
Example 3B Continued
Solve by completing the square.
Step 3
Complete the square.
Rewrite using like
denominators.
Step 4
Factor and simplify.
Step 5
Take the square root
of both sides.
Holt Algebra 1
9-8 Completing the Square
Example 3B Continued
Solve by completing the square.
Write and solve
two equations.
Step 6
The solutions are
Holt Algebra 1
and –4.
9-8 Completing the Square
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.
Holt Algebra 1
9-8 Completing the Square
Check It Out! Example 3a Continued
Solve by completing the square.
Step 2
.
Step 3
Complete the square.
Step 4
Factor and simplify.
Holt Algebra 1
9-8 Completing the Square
Check It Out! Example 3a Continued
Solve by completing the square.
Step 5
Take the square root
of both sides.
Step 6
Write and solve two
equations.
−
Holt Algebra 1
9-8 Completing the Square
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.
Holt Algebra 1
9-8 Completing the Square
Check It Out! Example 3b Continued
Solve by completing the square.
4t2 – 4t + 9 = 0
Step 2
.
Step 3
Complete the square.
Step 4
Factor and simplify.
There is no real number whose square is negative, so
there are no real solutions.
Holt Algebra 1
9-8 Completing the Square
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.
1
Understand the Problem
The answer will be the length and width of
the room.
List the important information:
• The room area is 195 square feet.
• The width is 2 feet less than the length.
Holt Algebra 1
9-8 Completing the Square
Example 4 Continued
2
Make a Plan
Set the formula for the area of a rectangle equal
to 195, the area of the room.
Solve the equation.
Holt Algebra 1
9-8 Completing the Square
Example 4 Continued
3
Solve
Let x be the width.
Then x + 2 is the length.
Use the formula for area of a rectangle.
l
•
w
length
times
width
x+2
Holt Algebra 1
•
x
=
A
= area of room
=
195
9-8 Completing the Square
Example 4 Continued
Step 1 x2 + 2x = 195
Step 2
Simplify.
.
Step 3 x2 + 2x + 1 = 195 + 1 Complete the square by
adding 1 to both sides.
Step 4 (x + 1)2 = 196
Factor the perfect-square
trinomial.
Take the square root of
Step 5 x + 1 = ± 14
both sides.
Step 6 x + 1 = 14 or x + 1 = –14 Write and solve two
equations.
x = 13 or x = –15
Holt Algebra 1
9-8 Completing the Square
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.
4
Look Back
The length of the room is 2 feet greater than the
width. Also 13(15) = 195.
Holt Algebra 1
9-8 Completing the Square
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.
1
Understand the Problem
The answer will be the length and width of
the room.
List the important information:
• The room area is 400 square feet.
• The length is 8 feet more than the width.
Holt Algebra 1
9-8 Completing the Square
Check It Out! Example 4 Continued
2
Make a Plan
Set the formula for the area of a rectangle equal
to 400, the area of the room.
Solve the equation.
Holt Algebra 1
9-8 Completing the Square
Check It Out! Example 4 Continued
3
Solve
Let x be the width.
Then x + 8 is the length.
Use the formula for area of a rectangle.
l
length
X+8
Holt Algebra 1
•
times
•
w
=
width
= area of room
x
=
A
400
9-8 Completing the Square
Check It Out! Example 4 Continued
Step 1 x2 + 8x = 400
Step 2
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 perfectsquare 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
x  16.4 or x  –24.4
equations.
Holt Algebra 1
9-8 Completing the Square
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.
4
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.
Holt Algebra 1
9-8 Completing the Square
Lesson Quiz: Part I
Complete the square to form a perfect square
trinomial.
1. x2 +11x +
2. x2 – 18x +
81
Solve by completing the square.
3. x2 – 2x – 1 = 0
4. 3x2 + 6x = 144
5. 4x2 + 44x = 23
Holt Algebra 1
6, –8
9-8 Completing the Square
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
Holt Algebra 1