# Completing the Square (Part 1)

ID: falal-fugoz
Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
PreviewAnswer Key

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 ﻿$x^{2} - 6x$﻿ in order to write it as a perfect square.

Show Work

2) Write an equivalent expression in factored form.

Show Work

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

Show Work

4) Write an equivalent expression in factored form.

Show Work

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

Show Work

6) Write an equivalent expression in factored form.

Show Work

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

Show Work

8) Write an equivalent expression in factored form.

Show Work

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

Show Work

10) Write an equivalent expression in factored form.

Show Work
##### Problem 2

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

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

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) ﻿$x^2 \ + \ 8x \ = \ 2$﻿

a)

(﻿$x$﻿ - 7﻿$)^{2}$﻿ = 54

b)

(﻿$x$﻿ + 5﻿$)^{2}$﻿ = 12

c)

(﻿$x$﻿ - 10﻿$)^{2}$﻿ = 91

d)

(﻿$x$﻿ + 4﻿$)^{2}$﻿ = 18

e)

(﻿$x$﻿ + 1﻿$)^{2}$﻿ = 1

f)

(﻿$x$﻿ + 2﻿$)^{2}$﻿ = 9

13) ﻿$x^2 \ + \ 10x \ = \ -13$﻿

a)

(﻿$x$﻿ - 7﻿$)^{2}$﻿ = 54

b)

(﻿$x$﻿ + 5﻿$)^{2}$﻿ = 12

c)

(﻿$x$﻿ - 10﻿$)^{2}$﻿ = 91

d)

(﻿$x$﻿ + 4﻿$)^{2}$﻿ = 18

e)

(﻿$x$﻿ + 1﻿$)^{2}$﻿ = 1

f)

(﻿$x$﻿ + 2﻿$)^{2}$﻿ = 9

14) ﻿$x^2 \ - \ 14x \ = \ 5$﻿

a)

(﻿$x$﻿ - 7﻿$)^{2}$﻿ = 54

b)

(﻿$x$﻿ + 5﻿$)^{2}$﻿ = 12

c)

(﻿$x$﻿ - 10﻿$)^{2}$﻿ = 91

d)

(﻿$x$﻿ + 4﻿$)^{2}$﻿ = 18

e)

(﻿$x$﻿ + 1﻿$)^{2}$﻿ = 1

f)

(﻿$x$﻿ + 2﻿$)^{2}$﻿ = 9

15) ﻿$x^2 \ + \ 2x \ = \ 0$﻿

a)

(﻿$x$﻿ - 7﻿$)^{2}$﻿ = 54

b)

(﻿$x$﻿ + 5﻿$)^{2}$﻿ = 12

c)

(﻿$x$﻿ - 10﻿$)^{2}$﻿ = 91

d)

(﻿$x$﻿ + 4﻿$)^{2}$﻿ = 18

e)

(﻿$x$﻿ + 1﻿$)^{2}$﻿ = 1

f)

(﻿$x$﻿ + 2﻿$)^{2}$﻿ = 9

16) ﻿$x^2 \ + \ 4x \ - \ 5 \ = \ 0$﻿

a)

(﻿$x$﻿ - 7﻿$)^{2}$﻿ = 54

b)

(﻿$x$﻿ + 5﻿$)^{2}$﻿ = 12

c)

(﻿$x$﻿ - 10﻿$)^{2}$﻿ = 91

d)

(﻿$x$﻿ + 4﻿$)^{2}$﻿ = 18

e)

(﻿$x$﻿ + 1﻿$)^{2}$﻿ = 1

f)

(﻿$x$﻿ + 2﻿$)^{2}$﻿ = 9

17) ﻿$x^2 \ - \ 20x \ = \ -9$﻿

a)

(﻿$x$﻿ - 7﻿$)^{2}$﻿ = 54

b)

(﻿$x$﻿ + 5﻿$)^{2}$﻿ = 12

c)

(﻿$x$﻿ - 10﻿$)^{2}$﻿ = 91

d)

(﻿$x$﻿ + 4﻿$)^{2}$﻿ = 18

e)

(﻿$x$﻿ + 1﻿$)^{2}$﻿ = 1

f)

(﻿$x$﻿ + 2﻿$)^{2}$﻿ = 9

##### Problem 4

Solve each equation by completing the square.

18) ﻿$x^{2} - 6x + 5 = 12$﻿

Show Work

19) ﻿$x^{2} - 2x = 8$﻿

Show Work

20) ﻿$11 = x^{2} + 4x -1$﻿

Show Work

21) ﻿$x^{2} - 18x + 60 = -21$﻿

Show Work
##### Problem 5

Rewrite each expression in standard form.

22) ﻿$(x + 3)(x - 3)$﻿

Show Work

23) ﻿$(7 + x)(x - 7)$﻿

Show Work

24) ﻿$(2x - 5)(2x + 5)$﻿

Show Work

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

Show Work
##### Problem 6

26) To find the product ﻿$203 \cdot 97$﻿ without a calculator, Priya wrote ﻿$(200 + 3)(200 - 3)$﻿. Very quickly, and without writing anything else, she arrived at ﻿$39,991$﻿. 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.

27) Select ﻿$\textbf{all}$﻿ 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?​​​​​​