Download Lesson 9.1 - Part 2

Chapter 9
Section 1
Part 2
9.1
Part 2 - Completing the Square
Objectives
1
Solve quadratic equations by completing the square.
2
Solve quadratic equations with solutions that are not real numbers.
3
4
5
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Solve quadratic equations by
completing the square.
There are times when you are unable to factor a trinomial in
order to solve an equation – or one side of the equation is
not in the form of a binomial squared. You can use a process
called Completing the Square.
Definition of Completing the Square: To manipulate one side
of the equation (and add that number to both sides) in order
to convert one side to be a perfect square trinomial that can
be factored as (x+k)^2 - a binomial squared.
For example – what would it take to turn x^2+6x +____
into a perfect square trinomial? (+9)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 3
Solve quadratic equations by completing the square.
Completing the Square
To solve ax2 + bx + c = 0 (a ≠ 0) by completing the square, use
these steps.
Step 1 Be sure the second-degree (squared) term has
coefficient 1. If the coefficient of the squared term is one,
proceed to Step 2. If the coefficient of the squared term is
not 1 but some other nonzero number a, divide each side of
the equation by a.
Step 2 Write the equation in correct form so that terms with
variables are on one side of the equals symbol and the
constant is on the other side.
Step 3 Square half the coefficient of the first-degree (linear)
term.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 4
Solve quadratic equations by completing the square.
Completing the Square (continued)
Step 4 Add the square to each side.
Step 5 Factor the perfect square trinomial. One side should now
be a perfect square trinomial. Factor it as the square of a
binomial. Simplify the other side.
Step 6 Solve the equation. Apply the square root property to
complete the solution.
Steps 1 and 2 can be done in either order. With some equations, it is more
convenient to do Step 2 first.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 5
CLASSROOM
EXAMPLE 1
Solving a Quadratic Equation by Completing the Square (a = 1)
Solve x2 – 2x – 10 = 0.
Solution:
x2 – 2x = 10
Completing the square – which means we need to find a number to add
to both sides of the equation that will make the left-hand side a perfect
2
square
1
2


trinomial:
2  1  1.
 2 


 
**In order to complete the square – you take ½ of the coefficient of the
middle term and square it, then you add it to both sides!**
So you add 1 to each side.
x2 – 2x + 1= 10 + 1
(x – 1)2 = 11
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 6
CLASSROOM
EXAMPLE 1
Solving a Quadratic Equation by Completing the Square (a = 1) (cont’d)
Use the square root property.
x 1  11
x  1  11
Check:
x 1   11
x  1  11
or
or
x  1  11 :
2
1  11  2 1  11  10  0




12  2 11  2  2 11 10  0
The solution set is
1 
?
True

11 .
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 7
CLASSROOM
EXAMPLE 2
Solving a Quadratic Equation by Completing the Square (a = 1)
Solve x2 + 3x – 1 = 0.
Solution:
x2 + 3x = 1
Completing the square.
2
2
1   3  9
 2  3    2   4
Add the square
to each side.
x2
9
9
+ 3x + = 1 +
4
4
2
3  13

x  
2
4

Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 8
CLASSROOM
EXAMPLE 2
Solving a Quadratic Equation by Completing the Square (a = 1) (cont’d)
Use the square root property.
3
13
x 
2
4
3
13
x 
2
2
3  13
x
2
Check that the solution set is
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
or
or
or
3
13
x 
2
4
3
13
x 
2
2
3  13
x
2
 3  13 

.
2


Slide 9.1- 9
CLASSROOM
EXAMPLE 3
Solving a Quadratic Equation by Completing the Square (a ≠ 1)
Solve 3x2 + 6x – 2 = 0.
Solution:
3x2 + 6x = 2
2
2
x  2x 
3
Completing the square.

1
2
 2
2
 1  1
2
2
x  2x 1  1
3
2
5
 x  1 
3
2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 10
CLASSROOM
EXAMPLE 3
Solving a Quadratic Equation by Completing the Square (a ≠ 1) (cont’d)
Use the square root property.
5
x 1 
3
5
x  1 
3
15
x  1 
3
3  15
x
3
Check that the solution set is
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
or
3
5
x 1  
2
3
or
5
x  1 
3
15
or
x  1 
3
3  15
or
x
3
 3  15 

.
3


Slide 9.1- 11
Objective 2
Solve quadratic equations with
solutions that are not real
numbers.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 12
CLASSROOM
EXAMPLE 4
Solve for Nonreal Complex Solutions
Solve the equation.
Solution:
x 2  17
x  17
or
x   17
x  i 17
or
x  i 17
The solution set is
i

17 .
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 13
CLASSROOM
EXAMPLE 4
Solve for Nonreal Complex Solutions (cont’d)
Solve the equation.
Solution:
 x  5
2
 100
x  5  100
or
x  5   100
x  5  10i
or
x  5  10i
x  5 10i
or
x  5 10i
The solution set is
5  10i.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 14
CLASSROOM
EXAMPLE 4
Solve for Nonreal Complex Solutions (cont’d)
Solve the equation.
Solution:
5 x 2  15 x  12  0
Complete the square.
5 x 2  15 x  12
12
2
x  3x  
5
2
 3 9
  3      
4
 2
9
12 9
2
x  3x    
4
5 4
2
3
3

x  
2
20

1
2
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2
Slide 9.1- 15
CLASSROOM
EXAMPLE 4
Solve for Nonreal Complex Solutions (cont’d)
3
3
x  
2
20
3 i 3 5
x 

2
20 5
3 i 15
x 
2
10
3
3
or x    
2
20
3 i 3 5
or x  

2
20
5
3 i 15
or x  
2
10
3 i 15
3 i 15
x 
or x  
2 10
2 10
 3
15 
The solution set is  
i .
 2 10 
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 16