10.2
Completing the Square
10.2
OBJECTIVES
1. Complete the square for a trinomial expression
2. Solve a quadratic equation by completing the square
We can solve a quadratic equation such as
x 2 2x 1 5
very easily if we notice that the expression on the left is a perfect-square trinomial.
Factoring, we have
(x 1)2 5
so
x 1 15
or
x 1 15
The solutions for the original equation are then 1 15 and 1 15.
It is true that every quadratic equation can be written in the form above (with a perfectsquare trinomial on the left). That is the basis for the completing-the-square method for
solving quadratic equations.
First, let’s look at two perfect-square trinomials.
x 2 6x 9 (x 3)2
(1)
x 2 8x 16 (x 4)2
(2)
There is an important relationship between the coefficient of the middle term (the x term)
and the constant.
In equation (1),
1
6
2
2
32 9
The x coefficient
The constant
In equation (2),
1
(8)
2
2
(4)2 16
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The x coefficient
The constant
It is always true that, in a perfect-square trinomial with a coefficient of 1 for x 2, the square
of one-half of the x coefficient is equal to the constant term.
Example 1
Completing the Square
(a) Find the term that should be added to x2 4x so that the expression is a
perfect-square trinomial.
761
762
CHAPTER 10 QUADRATIC EQUATIONS
NOTE The coefficient of x2
must be 1 before the added
term is found.
To complete the square of x2 4x, add the square of one-half of 4 (the x coefficient).
x2 4x 1
4
2
2
x2 4x 22
or
or
x2 4x 4
The trinomial x2 4x 4 is a perfect square because
x2 4x 4 (x 2)2
(b) Find the term that should be added to x2 10x so that the expression is a
perfect-square trinomial.
To complete the square of x2 10x, add the square of one-half of 10
(the x coefficient).
x2 10x 2 (10)
1
2
or
x2 10x (5)2
or
x2 10x 25
Check for yourself, by factoring, that this is a perfect-square trinomial.
CHECK YOURSELF 1
Complete the square and factor.
(a) x 2 2x
(b) x2 12x
We can now use the above process along with the solution methods of Section 10.1 to
solve a quadratic equation.
Example 2
Solving a Quadratic Equation by Completing the Square
Solve x2 4x 2 0 by completing the square.
NOTE Add 2 to both sides to
remove 2 from the left side.
x2 4x 2
We find the term needed to complete the square by squaring one-half of the x coefficient.
2 4
1
2
22 4
NOTE This completes the
square on the left.
x 2 4x 4 2 4
Now factor on the left and simplify on the right.
(x 2)2 6
Now solving as before, we have
x 2 16
x 2 16
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We now add 4 to both sides of the equation.
COMPLETING THE SQUARE
SECTION 10.2
763
CHECK YOURSELF 2
Solve by completing the square.
x 2 6x 4 0
For the completing-the-square method to work, the coefficient of x2 must be 1. Example 3 illustrates the solution process when the coefficient of x2 is not equal to 1.
Example 3
Solving a Quadratic Equation by Completing the Square
Solve 2x2 4x 5 0 by completing the square.
2x 2 4x 5 0
Add 5 to both sides.
2x 4x 5
2
x 2 2x x 2 2x 1 (x 1)2 5
2
5
1
2
NOTE
7
2
7
A2
A2 A2
x 1
14
114
A 4
2
Complete the square and
solve as before.
7
2
7
x 1
A2
Because the coefficient of x2 is not 1 (here it is 2), divide
every term by 2. This will make the new leading
coefficient equal to 1.
114
2
x1 Simplify the radical on the
right.
114
2
or
NOTE We have combined the
terms on the right with the
common denominator of 2.
x
2 114
2
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CHECK YOURSELF 3
Solve by completing the square.
3x2 6x 2 0
Let’s summarize by listing the steps to solve a quadratic equation by completing the
square.
CHAPTER 10 QUADRATIC EQUATIONS
Step by Step: Solving a Quadratic Equation by Completing
the Square
Step 1
Write the equation in the form
ax2 bx k
Step 2
Step 3
Step 4
so that the variable terms are on the left side and the constant is on
the right side.
If the coefficient of x 2 is not 1, divide both sides of the equation by
that coefficient.
Add the square of one-half the coefficient of x to both sides of the
equation.
The left side of the equation is now a perfect-square trinomial. Factor
and solve as before.
CHECK YOURSELF ANSWERS
1. (a) x 2 2x 1 (x 1)2; (b) x 2 12x 36 (x 6)2
3 13
2. 3 113
3.
3
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Name
10.2 Exercises
Section
Date
Determine whether each of the following trinomials is a perfect square.
1. x2 14x 49
2. x2 9x 16
3. x2 18x 81
4. x2 10x 25
5. x2 18x 81
6. x2 24x 48
Find the constant term that should be added to make each of the following expressions
a perfect-square trinomial.
7. x 6x
8. x 8x
9. x2 10x
10. x2 5x
2
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
2
16.
17.
11. x2 9x
12. x2 20x
18.
Solve each of the following quadratic equations by completing the square.
13. x2 4x 12 0
14. x2 6x 8 0
15. x2 2x 5 0
16. x2 4x 7 0
19.
20.
21.
22.
23.
17. x2 3x 27 0
18. x2 5x 3 0
24.
19. x2 6x 1 0
20. x2 4x 4 0
25.
21. x2 5x 6 0
22. x2 6x 3 0
23. x2 6x 5 0
24. x2 2x 1
26.
27.
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28.
25. x2 9x 5
26. x2 4 7x
27. 2x2 6x 1 0
28. 2x2 10x 11 0
30.
29. 2x2 4x 1 0
30. 2x2 8x 5 0
31.
31. 4x2 2x 1 0
32. 3x2 x 2 0
29.
32.
765
ANSWERS
33.
Solve each quadratic equation by completing the square.
34.
33. 3x2 4x 7x 9 2x2 5x 4
35.
34. 4x2 8x 4x 5 5x 2 2x 16
36.
37.
Solve the following problems.
38.
35. Number problem. If the square of 3 more than a number is 9, find the number(s).
a.
36. Number problem. If the square of 2 less than an integer is 16, find the number(s).
b.
37. Revenue. The revenue for selling x units of a product is given by R x 25 c.
Find the number of units sold if the revenue is $294.50.
1
x
2
d.
38. Number problem. Find two consecutive positive integers such that the sum of their
e.
squares is 85.
f.
Getting Ready for Section 10.3 [Section 1.5]
Evaluate the expression b2 4ac for each set of values.
(a) a 1, b 1, c 3
(c) a 1, b 8, c 3
(e) a 2, b 4, c 2
(b) a 1, b 1, c 1
(d) a 1, b 2, c 1
(f) a 2, b 3, c 4
Answers
1. Yes
3. No
15. 1 16
23. 3 114
1 15
4
b. 5
c. 76
17.
7. 9
3 3113
2
9 1101
25.
2
33. 1 16
d. 8
e. 0
9. 25
11.
81
4
19. 3 110
27.
3 17
2
35. 6, 0
13. 6, 2
21. 2, 3
29.
37. 19, 31
2 12
2
a. 13
f. 23
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31.
5. Yes
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