﻿ Solving a Quadratic Equation by Completing the Square Worksheet | Problems & Solutions

# Solving a Quadratic Equation by Completing the Square Worksheet

Solving a Quadratic Equation by Completing the Square Worksheet
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1.
What term should be added to the expression, $x$2 - 6$x$, to make it a perfect square trinomial?
 a. - 9 b. 36 c. 9 d. None of the above

#### Solution:

x2 - 6x
[Original expression.]

To complete the square of the expression x2 + bx, add the square of half the coefficient of x, that is, add ( b / 2)2 .

The coefficient b = -6, so add ( b / 2)2 = (-3)2, to the expression.

x2 - 6x + (-3)2 = x2 - 6x + 9 = (x - 3)2 .

9 should be added to the expression, x2 - 6x, to create a perfect square trinomial.

2.
Josh wants to mark a rectangular plot of area 20900 ft2 for planting orange trees. What would be the dimensions of the plot, if he wants the length to be 80 feet more than the width?
 a. 100 ft by 200 ft b. 150 ft by 140 ft c. 110 ft by 190 ft d. None of the above

#### 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
[To make the RHS a perfect square, 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

The plot will be 190 ft long and 110 ft wide.

3.
What term should be added to the expression, $x$2 - 18$x$, to create a perfect square trinomial?
 a. 81 b. 18 c. -81 d. None of the above

#### Solution:

x2 - 18x
[Original expression.]

To complete the square of the expression x2 + bx, add the square of half the coefficient of x, that is, add ( b / 2)2

The coefficient of x is - 18, so add (- 18 / 2)2 = (-9)2 , to the expression.

x2 - 18x + (-9)2 = x2 - 18x + 81 = (x - 9)2.

81 should be added to the expression x2 - 18x, to create a perfect square trinomial.

4.
What term should be added to the expression, $x$2 + 18$x$, to create a perfect square trinomial?
 a. - 81 b. 81 c. 18 d. None of the above

#### Solution:

x2 + 18x
[Original expression.]

To complete the square of the expression x2 + bx, add the square of half the coefficient of x, that is, add (b/2)2

The coefficient of x is 18, so add (18 / 2)2 = 92 to the expression.

x2 + 18x + (9)2 = x2 + 18x + 81 = (x + 9)2

81 should be added to the expression x2 + 18x, to create a perfect square trinomial.

5.
Solve $x$2 + 4$x$ = 21 by completing the square.
 a. -7 and 3 b. -7 and -3 c. 1 and -21 d. None of the above

#### Solution:

x2 + 4x = 21
[Original equation.]

x2 + 4x + 22 = 21 + 22
[Add ( 4 / 2)2 = 22 to each side.]

(x + 2)2 = 25
[Writing left side as perfect square.]

(x + 2) = ± 5
[Finding square roots on each side.]

x = - 2 ± 5
[Subtract 2 from each side.]

x = -7 or x = 3
[Simplify.]

The solutions of the equation x2 + 4x = 21 are -7 and 3.

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

#### 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.

7.
The area of the right triangle is 70 square cm. What is the value of $x$?

 a. 10 b. 8 c. 15 d. 12

#### Solution:

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

70 = 12 x (x + 4) x x
[Substitute 70 for A, x for height and (x + 4) for base.]

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

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

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

140 + 22 = x2 + 4x + 22
[To make RHS a perfect square, add (4/2)2 = (2)2 to each side.]

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

± 12 = (x + 2)
[Find square roots on each side.]

x = ±12 - 2
[Subtract 2 from each side.]

x = 10
[Since x is the height of the triangle, discard the negative value.]

8.
The area of the parallelogram 40 cm2. What is its length?

 a. 7 cm b. 9 cm c. 10 cm d. 8 cm

#### Solution:

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

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

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

40 + 32 = x2 + 6x + 32
[Add ( 6 / 2)2 = (3)2 = 9 to each side.]

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

± 7 = (x + 3)
[Evaluate square roots on both sides.]

x = ± 7 - 3
[Subtract 3 from each side.]

x = 4 or -10
[Simplify.]

Height of the parallelogram = x = 4 cm
[Since height cannot be a negative value.]

Length = x + 6 = 4 + 6 = 10 cm
[Simplify.]

9.
Mr. Jim is planning to have a garden of 169 square feet. What are the dimensions of the garden, if he wants it in the form of a square?
 a. 5 feet by 5 feet b. 13 feet by 13 feet c. 8 feet by 8 feet d. 20 feet by 20 feet

#### Solution:

Let x be the side of the garden, Mr.Jim is planning to have.

The area of a square = (side)2.

x2 = 169
[Original equation.]

x = ± 13
[Find square roots on each side.]

The dimensions of the garden would be 13 feet by 13 feet.
[Dimension cannot be a negative value.]

10.
Joe's apartment complex has a rectangular skating floor. The floor is $x$ ft long and ($x$ + 10) ft wide. What are the dimensions of the skating floor, if its area is 600 ft2?
 a. 22 ft by 30 ft b. 20 ft by 25 ft c. 20 ft by 30 ft d. 20 ft by 33 ft

#### Solution:

The area of a rectangle = width x length

600 = x(x + 10)
[Original equation.]

600 = x2 + 10x
[Use distributive property to simplify.]

600 + 52 = x2 + 10x + 52
[Add (10 / 2)2 = 52 on each side.]

625 = (x + 5)2
[Write right side as a perfect square and simplify.]

±25 = (x + 5)
[Evaluate square roots on both sides.]

x = ±25 - 5
[Subtract 5 from each side.]

x = 20 or -30
[Simplify.]

The skating floor is x = 20 ft long and (x + 10) = 30 ft wide.
[Since x is the length of the floor, consider positive value.]