Completing the Square (Part 1)

ID: falal-fugoz
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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28 questions

Completing the Square (Part 1)

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Problem 1

Add the number that would make the expression a perfect square. Next, write an equivalent expression in factored form.

1) Add a number to x26xx^{2} - 6x in order to write it as a perfect square.

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2) Write an equivalent expression in factored form.

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3) Add a number to x2+2xx^{2} + 2x in order to write it as a perfect square.

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4) Write an equivalent expression in factored form.

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5) Add a number to x2+14xx^{2} + 14x in order to write it as a perfect square.

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6) Write an equivalent expression in factored form.

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7) Add a number to x24xx^{2} - 4x in order to write it as a perfect square.

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8) Write an equivalent expression in factored form.

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9) Add a number to x2+24xx^{2} + 24x in order to write it as a perfect square.

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10) Write an equivalent expression in factored form.

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Problem 2

11) Mai is solving the equation x2+12x=13x^{2} + 12x = 13. She writes:

x2+12x=13(x+6)2=49x=1orx=13\begin{array}{rcl} \\[-1em] x^{2} + 12x &=& 13 \\[-1em] \\ \\[-1em] (x + 6)^{2} &=& 49 \\[-1em] \\ \\[-1em] x = 1 &\text{or}& x = -13 \\[-1em] \end{array}

Jada looks at Mai’s work and is confused. She doesn’t see how Mai got her answer.

Complete Mai’s missing steps to help Jada see how Mai solved the equation.

Problem 3

Match each equation to an equivalent equation with a perfect square on one side.

12) x2 + 8x = 2x^2 \ + \ 8x \ = \ 2

a)

(xx - 7)2)^{2} = 54

b)

(xx + 5)2)^{2} = 12

c)

(xx - 10)2)^{2} = 91

d)

(xx + 4)2)^{2} = 18

e)

(xx + 1)2)^{2} = 1

f)

(xx + 2)2)^{2} = 9

13) x2 + 10x = 13x^2 \ + \ 10x \ = \ -13

a)

(xx - 7)2)^{2} = 54

b)

(xx + 5)2)^{2} = 12

c)

(xx - 10)2)^{2} = 91

d)

(xx + 4)2)^{2} = 18

e)

(xx + 1)2)^{2} = 1

f)

(xx + 2)2)^{2} = 9

14) x2  14x = 5x^2 \ - \ 14x \ = \ 5

a)

(xx - 7)2)^{2} = 54

b)

(xx + 5)2)^{2} = 12

c)

(xx - 10)2)^{2} = 91

d)

(xx + 4)2)^{2} = 18

e)

(xx + 1)2)^{2} = 1

f)

(xx + 2)2)^{2} = 9

15) x2 + 2x = 0x^2 \ + \ 2x \ = \ 0

a)

(xx - 7)2)^{2} = 54

b)

(xx + 5)2)^{2} = 12

c)

(xx - 10)2)^{2} = 91

d)

(xx + 4)2)^{2} = 18

e)

(xx + 1)2)^{2} = 1

f)

(xx + 2)2)^{2} = 9

16) x2 + 4x  5 = 0x^2 \ + \ 4x \ - \ 5 \ = \ 0

a)

(xx - 7)2)^{2} = 54

b)

(xx + 5)2)^{2} = 12

c)

(xx - 10)2)^{2} = 91

d)

(xx + 4)2)^{2} = 18

e)

(xx + 1)2)^{2} = 1

f)

(xx + 2)2)^{2} = 9

17) x2  20x = 9x^2 \ - \ 20x \ = \ -9

a)

(xx - 7)2)^{2} = 54

b)

(xx + 5)2)^{2} = 12

c)

(xx - 10)2)^{2} = 91

d)

(xx + 4)2)^{2} = 18

e)

(xx + 1)2)^{2} = 1

f)

(xx + 2)2)^{2} = 9

Problem 4

Solve each equation by completing the square.

18) x26x+5=12x^{2} - 6x + 5 = 12

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19) x22x=8x^{2} - 2x = 8

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20) 11=x2+4x111 = x^{2} + 4x -1

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21) x218x+60=21x^{2} - 18x + 60 = -21

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Problem 5

Rewrite each expression in standard form.

22) (x+3)(x3)(x + 3)(x - 3)

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23) (7+x)(x7)(7 + x)(x - 7)

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24) (2x5)(2x+5)(2x - 5)(2x + 5)

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25) (x+18)(x18)(x + \frac{1}{8})(x - \frac{1}{8})

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Problem 6

26) To find the product 20397203 \cdot 97 without a calculator, Priya wrote (200+3)(2003)(200 + 3)(200 - 3). Very quickly, and without writing anything else, she arrived at 39,99139,991. Explain how writing the two factors as a sum and a difference may have helped Priya.

Problem 7

A basketball is dropped from the roof of a building and its height in feet is modeled by the function h.

Here is a graph representing h.

A template for answering this question. Ask your instructor for an alternative.

27) Select all\textbf{all} the true statements about this situation. Write each corresponding letter in the answer box and separate letters with commas.

a) When t=0 the height is 0 feet.

b) The basketball falls at a constant speed.

c) The expression that defines h is linear.

d) The expression that defines h is quadratic.

e) When t=0 the ball is about 50 feet above the ground.

f) The basketball lands on the ground about 1.75 seconds after it is dropped.

Problem 8

28) A group of students are guessing the number of paper clips in a small box.

The guesses and the guessing errors are plotted on a coordinate plane.

What is the actual number of paper clips in the box?​​​​​​

A template for answering this question. Ask your instructor for an alternative.