AnswersAlgebra III ASolving Quadratic Equations: Completing the Square

Solving Quadratic Equations: Completing the Square — Unit test Answers

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Which function has real zeros at x = −10 and x = −6?

f
f(x) = x2 + 16x + 60
f
f(x) = x2 − 16x + 60
f
f(x) = x2 + 4x + 60
f
f(x) = x2 − 4x + 60
2
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An architect designs a diagonal path across a rectangular patio. The path is 29 meters long. The width of the patio is x meters, and the length of the path is 5 meters more than the width.

Question illustration
A
0.5(x)(x + 5) = 29
B
0.5(x)(x + 5) = 841
C
x2 + (x + 5)2 = 29
D
x2 + (x + 5)2 = 841
3

Solve x2 − 24x = −80 by completing the square.What is the solution set of the equation?

A
{2, 40}
B
{4, 20}
C
{5, 16}
D
{8, 10}
4

Which equation has an a-value of –2, a b-value of 1, and a c-value of 3?

A
0 = –2x2 + x + 3
B
0 = 2x2 + x + 3
C
0 = –2x2 + 3
D
0 = 2x2 – x + 3
5

The first two steps in determining the solution set of the system of equations, y = –x2 + 4x + 12 and y = –3x + 24, algebraically are shown in the table. Which represents the solution(s) of this system of equations?

Question illustration
A
(3, 15)
B
(–4, 36)
C
(3, 15) and (4, 12)
D
(–3, 33) and (–4, 36)
6

The product of two consecutive integers is 272. Which quadratic equation can be used to find x, the smaller number?

A
x2 + 1 = 272
B
x2 − 1 = 272
C
x2 + x − 272 = 0
D
x2 + x + 272 = 0
7

Clara writes the equation (x – 13)(x + 8) = 196 to solve for the missing side length of a rectangle represented by the factor x + 8. What is the missing side length represented by x + 8 units of the rectangle?

A
7
B
15
C
20
D
28
8

Two numbers are 10 units away in different directions from their midpoint, m, on a number line. The product of the numbers is –99.

A
(m – 5)(m + 5) = 99
B
(m – 10)(m + 10) = 99
C
m2 – 25 = –99
D
m2 – 100 = –99
9

Between which two ordered pairs does the graph of f(x) = x2 + x – 9 cross the negative x-axis?Quadratic formula: x =

Question illustration
A
(–6, 0) and (–5, 0)
B
(–4, 0) and (–3, 0)
C
(–3, 0) and (–2, 0)
D
(–2, 0) and (–1, 0)
10

Two sandboxes with the same area are shown.The equation represents the area of Sandbox 2 in terms of its width.

Question illustration
A
2.72 meters
B
3.06 meters
C
9.16 meters
D
10.18 meters
11

A rectangle’s length is 5 inches more than twice its width. Its area is 50 square inches. Which equation can be used to find its width, w?

A
2w(2w + 5) = 50
B
2w(w + 5) = 50
C
w(2w + 5) = 50
D
w(w + 5) = 50
12

What values of b satisfy 4(3b + 2)2 = 64?

A
b = and b = –2
Option A
B
b = 2 and b =
Option B
C
b = and b = 3
Option C
D
b = 2 and b =
Option D
13

Liliana decides to crop a square photo 2 inches on each side to fit it into a frame. The area of the original photo was 121 square inches. In the equation (x + 2)2 = 121, x represents the side measure of the cropped photo.

A
8 inches by 8 inches
B
9 inches by 9 inches
C
12 inches by 12 inches
D
13 inches by 13 inches
14

What are the solutions to the quadratic equation 4(x + 2)2 = 36

A
x = −11 and x = 7
B
x = −7 and x = 11
C
x = −5 and x = 1
D
x = −1 and x = 5
15

What is a solution to (x + 6)(x + 2) = 60?

A
x = −6
B
x = −4
C
x = 4
D
x = 12
16

Which is the graph of a quadratic equation that has a negative discriminant?

A
On a coordinate plane, a parabola opens down. It goes through (negative 4, negative 4), has a vertex at (negative 2, 0), and goes through (0, negative 4).
Option A
B
On a coordinate plane, a parabola opens down. It goes through (negative 4, negative 4), has a vertex at (negative 2, 0), and goes through (0, negative 4).
Option B
C
On a coordinate plane, a parabola opens up. It goes through (negative 4, 4), has a vertex at (negative 2, 0), and goes through (0, 4).
Option C
D
On a coordinate plane, a parabola opens up. It goes through (negative 4, 8), has a vertex at (negative 2, 4), and goes through (0, 8).
Option D
17

A quadratic equation of the form 0 = ax2 + bx + c has one real number solution. Which could be the equation?

A
0 = 2x2 – 4x + 1
B
0 = 2x2 – 5x + 3
C
0 = –2x2 – 4x – 2
D
0 = –2x2 – 3x – 1
18

The first three steps in determining the solution set of the system of equations algebraically are shown in the table.y = −x2 +2x − 9 y = −6x + 6

Question illustration
A
(5, −24) and (3, −12)
B
(5, 36) and (3, 24)
C
(−5, −24) and (−3, 12)
D
(−5, 36) and (−3, 24)
19

Solve x2 – 8x = 20 by completing the square. Which is the solution set of the equation?

A
{–22, 14}
B
{–14, 22}
C
{–10, 2}
D
{–2, 10}
20

Two numbers are 10 units away in different directions from their midpoint, m, on a number line. The product of the numbers is –99.

A
(m – 5)(m + 5) = 99
B
(m – 10)(m + 10) = 99
C
m2 – 25 = –99
D
m2 – 100 = –99

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Solving Quadratic Equations: Completing the Square