# Solve the Equation by Completing the Square Worksheet

Solve the Equation by Completing the Square Worksheet
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1.
Solve $x$2 + 10$x$ = 39 by completing the square.
 a. 1 and - 39 b. 10 and 39 c. - 13 and - 3 d. - 13 and 3

#### Solution:

x2 + 10x = 39
[Original equation.]

x2 + 10x + 52 = 39 + 52
[Add ( 10 / 2)2 = 52 to each side.]

(x + 5)2 = 64
[Writing left side as perfect square.]

(x + 5) = ± 8
[Finding square roots on each side.]

x = - 5 ± 8
[Subtract 5 from each side.]

x = - 13 or x = 3
[Simplify.]

The solutions of the equation x2 + 10x = 39 are - 13 and 3.

2.
Solve $x$2 - 6$x$ = 27 by completing the square.
 a. 11 and - 3 b. - 9 and - 3 c. 9 and - 7 d. 9 and - 3

#### Solution:

x2 - 6x = 27
[Original equation.]

x2 - 6x + 9 = 27 + 9
[Add (- 6 / 2)2 = (- 3)2 = 9 to each side.]

(x - 3)2 = 36
[Write left side as perfect square and simplify.]

x - 3 = ± 6
[Evaluate square roots on both sides.]

x = 3 ± 6

x = 9 or x = - 3
[Simplify.]

The solutions of the equation x2 - 6x = 27 are 9 and - 3.

3.
The area of the right triangle shown is 96 square cm. Find the value of $x$.

 a. 10 b. 14 c. 12 d. 17

#### Solution:

The area of a right triangle is given by, A = 1 / 2× base × height.

96 = 12 × (x + 4) × x
[Substitute A = 96, height = x and base = (x + 4).]

96 × 2 = 12× (x + 4) × x × 2
[Multiply each side by 2.]

192 = x(x + 4)
[Simplify.]

192 = x2 + 4x
[Use distributive property.]

192 + 22 = x2 + 4x + 22
[To make RHS a perfect square, add (4 / 2)2 = (2)2 to each side.]

196 = (x + 2)2
[Write right side as a perfect square.]

± 14 = (x + 2)
[Find square roots on each side.]

x = ±14 - 2
[Subtract 2 from each side.]

x = 12
[Since x is the height of the triangle, discard the negative value.]

4.
The area of the parallelogram is 16 cm2. What is its length?

 a. 8 cm b. 7 cm c. 5 cm d. 6 cm

#### Solution:

The area of a parallelogram is equal to the product of its height and length.

16 = x(x + 6)
[Original equation.]

16 = x2 + 6x
[Use distributive property to simplify.]

16 + 32 = x2 + 6x + 32
[Add (6 / 2)2 = (3)2 = 9 to each side.]

25 = (x + 3)2
[Write right side as a perfect square and simplify.]

± 5 = (x + 3)
[Evaluate square roots on both sides.]

x = ± 5 - 3
[Subtract 3 from each side.]

x = 2 or - 8
[Simplify.]

Height of the parallelogram is x = 2 cm.
[Since height cannot be a negative value.]

Length = x + 6 = 2 + 6 = 8 cm
[Simplify.]

5.
What term should be added to the expression $x$2 - 4$x$, to create a perfect square trinomial?
 a. - 16 b. - 4 c. 16 d. 4

#### Solution:

x2 - 4x
[Original expression.]

Add to the expression x2 + bx, the square of half the coefficient of x, that is (b2)2 to create a perfect square trinomial.

The coefficient b = - 4, so add (b2)2 = (- 2)2, to the expression.

x2 - 4x + (- 2)2 = x2 - 4x + 4 = (x - 2)2 .

4 should be added to the expression x2 - 4x, to create a perfect square trinomial.

6.
George wants to fence a rectangular plot. The area of the plot is 20900 ft2. Find the dimensions of the plot if the length is 80 feet more than the width.
 a. 200 ft by 120 ft b. 190 ft by 110 ft c. 150 ft by 70 ft d. 250 ft by 170 ft

#### Solution:

Let x and x + 80 be the width and length of the rectangular plot.

20900 = x(x + 80)
[Write an equation.]

20900 = x2 + 80x
[Use distributive property.]

20900 + 402 = x2 + 80x + 402
[Add ( 80 / 2)2 = (40)2 to each side.]

22500 = (x + 40)2
[Write the right side expression as a perfect square.]

± 150 = (x + 40)
[Apply square root on each side.]

x = ±150 - 40
[Subtract 40 from each side.]

x = 110 feet
[Discard the negative value of x.]

x + 80 = 110 + 80 = 190 ft

So, the plot will be 190 ft long and 110 ft wide.

7.
What term should be added to the expression, $x$2 - 12$x$ to create a perfect square trinomial?
 a. - 36 b. 6 c. 36 d. 1

#### Solution:

x2 - 12x
[Original expression.]

Add to the expression x2 + bx, the square of half the coefficient of x, that is (b2)2 to create a perfect square trinomial.

The coefficient of x is - 12, so add (- 12 / 2)2 = (- 6)2 , to the expression.

x2 - 12x + (- 6)2 = x2 - 12x + 36 = (x - 6)2.

36 should be added to the expression x2 - 12x, to create a perfect square trinomial.

8.
What term should be added to the expression, $x$2 + 4$x$ to create a perfect square trinomial?
 a. 5 b. 4 c. - 4 d. 2

#### Solution:

x2 + 4x
[Original expression.]

Add to the expression x2 + bx, the square of half the coefficient of x, that is (b2)² to create a perfect square trinomial.

The coefficient of x is 4, so add (4 / 2)2 = 22 to the expression.

x2 + 4x + (2)2 = x2 + 4x + 4 = (x + 2)2

4 should be added to the expression x2 + 4x, to create a perfect square trinomial.

9.
Charles's apartment complex has a rectangular skating floor. The floor is $x$ ft long and ($x$ + 6) ft wide. What are the dimensions of the skating floor, if its area is 616 ft2?
 a. 22 ft by 23 ft b. 22 ft by 31 ft c. 24 ft by 28 ft d. 22 ft by 28 ft

#### Solution:

The area of a rectangle = Width × Length

616 = x(x + 6)
[Original equation.]

616 = x2 + 6x
[Use distributive property to simplify.]

616 + 32 = x2 + 6x + 32
[Add (6 / 2)2 = 32 on each side.]

625 = (x + 3)2
[Write right side as a perfect square and simplify.]

± 25 = (x + 3)
[Evaluate square roots on both sides.]

x = ± 25 - 3
[Subtract 3 from each side.]

x = 22 or - 28
[Simplify.]

The skating floor is x = 22 ft long and (x + 6) = 28 ft wide.
[Since x is the length of the floor, consider positive value.]

10.
A rectangular painting is $a$ inches wide and $a$ + 8 inches long. What are the dimensions of the painting , if the area of the painting is 128 inches2?
 a. 10 inches by 18 inches b. 16 inches by 8 inches c. 9 inches by 17 inches d. 11 inches by 19 inches

#### Solution:

The area of a rectangle = Length × Width

The area of the rectangular painting = (a + 8) × (a) inches2

128 = (a + 8) × (a)
[Original equation.]

128 = a2 + 8a
[Use distributive property.]

128 + 42 = a2 + 8a + 42
[Add (8 / 2)2 = 42 = 16 to each side.]

144 = (a + 4)2
[Write the right side of the equation as a perfect square and simplify.]

± 12 = (a + 4)
[Evaluate square roots on both the sides.]

± 12 - 4 = a + 4 - 4
[Subtract 4 from each side.]

a = 8 or - 16
[Simplify.]

Width of the rectangular painting is a = 8 inches.
[Since the dimensions cannot be negative.]

Length of the rectangular painting is (a + 8) = (8 + 8) = 16 inches.
[Replace a with 8 and add.]

The dimensions of the rectangular painting are 16 inches by 8 inches.