UNPACKING the TEKS | R€¦ · [TEKS A.8(A), DOK 2] 5 What are the possible solutions to x2 2 x 4...

Solve Quadratic Equations by Completing the Square and Taking Square Roots

Algebra I | End-of-Coursemasteryeducation.com206

LESSON 35CHAPTER 5

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GUIDED PRACTICE

Word to Knowcompleting the square

Solve x2 � 6x � 7 � 0 for x by completing the square.Problem 1

UNPACKING the TEKS | R A.8(A)

Previously, you solved the quadratic equation (x � 4)2 � 5 by taking the square root of each side. But often a quadratic equation isn’t in a form where you can take the square root of both sides. In this case, you can use a method called completing the square to get the equation into this form.

Step 1 Move the constant to the right side of the equation.

Add 7 to both sides of the equation: x2 � 6x � .

Step 1 Take half of the coefficient of x, square it, and add it to both sides.

Half of 6 is 3. Squaring the 3 is .

Add 9 to both sides: x2 � 6x � 9 � 7 � .

Step 1 Write the left side as a perfect square and simplify the right side.

x2 � 6x � 9 � 7 � 9

( )2 � 16

Step 1 Solve for x.

(x � 3)2 � 16

x � 3 �

x � �3 4

x � or

Step 1

Step 2

Step 3

Step 4

So the solutions of x2 � 6x � 7 � 0 by completing the square are

x � or .Solution

9781609794408_TX_ALG_EOC_SE_Book_8-2018.indb 2069781609794408_TX_ALG_EOC_SE_Book_8-2018.indb 206 8/27/2018 5:09:27 PM8/27/2018 5:09:27 PMProcess BlackProcess Black

SAMPLE

masteryeducation.comChapter 5 | Quadratic Equations

LESSON 35 • SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE AND TAKING SQUARE ROOTS

207Copying is prohibited.

So the solutions of x2 � 8x � 6 � 10 by completing the square are

x � or x � .Solution

Solve 3x2 � 12x � 8 for x by completing the square.Problem 3

Factor a 3 from the left side of the equation.

3(x2 � ) � 8

Complete the square.

3(x2 � 4x � ) � 8 � 3( )

Since the 3 is multiplied by each term on the left, add 3 times 4 on the right.

3(x – 2)2 �

(x � 2)2 �

� √ __

20 ___ 3

x �

Step 1

Step 2

So the solutions to 3x2 � 12x � 8 by completing the square are x � 2 � √___20___3 .Solution

Solve x2 � 8x � 6 � 10 for x by completing the square.Problem 2

Step 1 Gather all constants on the right side of the equation.

x2 � 8x � 6 � 10

x2 � 8x �

Step 1 Take half of the coefficient of x, square it, and add it to both sides.

x2 � 8x � 4

x2 � 8x � � 4 �

(x � )2 � 20

Step 1 Write the left side as a perfect square, then solve for x.

(x � 4)2 � 20

x � 4 �

x �

Step 1

Step 2

Step 3


SAMPLE

x LESSON 35 • SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE AND TAKING SQUARE ROOTS

Algebra I | End-of-Coursemasteryeducation.com208 Copying is prohibited.

SHORT RESPONSE QUESTIONS1. Try to solve x2 � 8x � 18 � 0 by completing the square. Explain how you know there is no real

solution to this equation.

2. Eric solved an equation by completing the square.

5x2 � 4x � 2 � 0

5x2 � 4x � 2

5(x2 � 0.8x � 0.16) � 2 � 5(0.16)

5(x � 0.4)2 � 1.2

(x � 0.4)2 � 1.2 ___ 5 � 0.24

x � 0.4 � √ ____

0.24

x � 0.4 √ ____

0.24

Part A Find Eric’s error. Explain what he did incorrectly.

Part B Solve the equation correctly by completing the square.


SAMPLE

masteryeducation.comChapter 5 | Quadratic Equations

LESSON 35 • SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE AND TAKING SQUARE ROOTS

209Copying is prohibited.

PRACTICE

1 Given x2 � 4x � 7 � 12, what are the possible values for x?

A �5, 1

B 5, �1

C �5, �1

D 5, 1

[TEKS A.8(A), DOK 2]

Use the information for Problems 2–3.

Kim starts to solve 3x2 � 9x � 4 � 0 for x by completing the square.

3x2 � 9x � 4

3(x2 � 3x � A) � 4 � B

2 What does A equal?

F 1.5

G 2.25

H 6.75

J 9


3 What does B equal?

A 1.5

B 2.25

C 6.75

D 9


4 Joe solves 3x2 � 3x � 18 � 0 for x by completing the square. Which would be the final step of his solution?

F 0.5 √ _____

6.25

G �0.5 √ _____

6.25

H 1 √ __

7

J �1 √ __

7


5 What are the possible solutions to x2 � 2x � 4 � 12?

A x � 1 or x � 3

B x � 1 or x � �3

C x � �2 or x � 4

D x � 2 or x � 4


6 Given x2 � 10x � 2 � 4, what are the possible values for x?

F �5 √ ___

31

G 5 √ ___

31

H �1, 11

J 1, � 11


DIRECTIONS Read each question and choose the best answer.Then circle the letter for the correct answer.


SAMPLE

Algebra I | End-of-Coursemasteryeducation.com210

x LESSON 35 • SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE AND TAKING SQUARE ROOTS

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ASSESSMENT

1 Given x2 � 10x � 19 � �18, what are the possible values of x?

A 5, �5

B 2 √ __

3 5

C �2 √ __

3 5

D no real solutions


Use the information for Problems 2–3.

Larry starts to solve 5x2 � 40x � 15 � 0 for x by completing the square.

5(x2 � 8x) � �15

5(x2 � 8x � 16) � �15 � 16

2 What was his error?

F He didn’t recognize that there is no solution because the right side is �15.

G He added 16, which is 8(2) within the parentheses.

H He subtracted 15 from each side before factoring.

J He added 16 instead of 80 to the right side.


3 What is the correct solution?

A �4 √ ___

65

B 4 √ ___

65

C �4 √ ___

13

D 4 √ ___

13


4 Paul solves the equation 3x2 � 6x � 24 � 0 for x by completing the square. Which would be a valid step in the solution to the problem?

F (x � 1)2 � 8

G (x � 1)2 � 9

H (x � 1)2 � 9

J (x � 1)2 � 9(3)

[TEKS A.1(B), A.8(A), DOK 2]

5 Theo solved x2 � 8x � 7 � 2 for x by completing the square. He said the solution was 1. Which solution did he not find?

Record your answer and fill in the bubbles on the grid below. Be sure to use the correct place value.


DIRECTIONS Read each question and choose the best answer.Then circle the letter for the correct answer.


SAMPLE

UNPACKING the TEKS | R€¦ · [TEKS A.8(A), DOK 2] 5 What are the possible solutions to x2 2 x 4...

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