MATH 80
UNIT 8.1
SUPPLEMENT
SOLVING QUADRATIC EQUATIONS
BY COMPLETING THE SQUARE
When factoring, root extraction cannot be used to solve a quadratic equation; another method can be
employed called completing the square.
Example 1
.
Consider the equation
1. Isolate the x-terms by adding 9 to both sides of the equation.
2. Take of the coefficient in front of the x-terms and square it.
(
)
( )
=
3. Add this constant to both sides of the equation.
4. Left-hand side should be the perfect square of a binomial.
(
)
REMARK: The constant in the binomial will be half the coefficient of the -term.
5. Combine the terms on the right side of the equation.
6
(
)
(
)
Take the square root from both sides.
√
√
7. Solve for
by subtracting from both sides.
√
Therefore, the solution set is: {
√
or
√
√
} or {
√
}
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No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
MATH 8
Example 2
UNIT 8.1
Consider the equation
PAGE 2
.
1. Rearrange the -terms in descending order. The constant is already on the other side.
2
Divide both sides by 3. It is necessary that the coefficient in front of the x-term be a positive 1.
3. Take of the middle term and square it.
( )]
[
( )
4. Add this constant to both sides of the equation and simplify.
5. Factor the left-hand side.
(
)
6. Take the square root of both sides.
√
√
7. Solve for x by adding to both sides.
√
Therefore, the solution set is: {
√
or
√
}
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
MATH 80
Example 3
UNIT 8.1
Solve
PAGE 3
.
Subtract 7 from both sides
Divide by -4
Add [ (
)]
( )
(
)
Factor the left side and combine
fractions on right side
(
)
The square root of both sides and
Simplify the denominator
Solve for 𝑥
√
√
√
The solutions are: {
Problem Set A
}
Supply the missing term needed to complete the square.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Answers A
1.
2.
3.
4.
5.
6.
7.
8.
9.
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No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
10.
MATH 80
UNIT 8.1
Problem Set B
PAGE 4
Solve by completing the square.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Answers B
1. {
4. {
√
2. {
}
5. {
√
7. {
10. {
}
}
}
}
√
}
6.
{
}
9.
{
}
11. {
}
12.
{
}
15.
{
18.
{
14. {
}
16. {
}
17. {
√
}
}
}
}
√
{
8. {
13. {
19. {
√
3.
20. {
√
}
√
}
√
}
}
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.