AnswersAlgebra III ASolving Quadratic Equations: Completing the Square

Introduction to the Quadratic Formula Answers

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1
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Which shows the correct substitution of the values a, b, and c from the equation 1 = –2x + 3x2 + 1 into the quadratic formula?Quadratic formula:

Question illustration
A
x = StartFraction negative (negative 2) plus or minus StartRoot (negative 2) squared minus 4 (3)(0) EndRoot Over 2(3) EndFraction
Option A
B
x = StartFraction negative (negative 2) plus or minus StartRoot (negative 2) squared minus 4 (3)(2) EndRoot Over 2(3) EndFraction
Option B
C
x = StartFraction negative (negative 2) plus or minus StartRoot (negative 2) squared minus 4 (3)(1) EndRoot Over 2(3) EndFraction
Option C
D
x = StartFraction negative 3 plus or minus StartRoot 3 squared minus 4 (negative 2)(0) EndRoot Over 2(negative 2) EndFraction
Option D
2
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What is the value of the discriminant, b2 − 4ac, for the quadratic equation 0 = x2 − 4x + 5, and what does it mean about the number of real solutions the equation has?

A
The discriminant is −4, so the equation has 2 real solutions.
B
The discriminant is −4, so the equation has no real solutions.
C
The discriminant is 35, so the equation has 2 real solutions.
D
The discriminant is 35, so the equation has no real solutions.
3

A student is deriving the quadratic formula. Her first two steps are shown.Step 1: –c = ax2 + bxStep 2:

Question illustration
A
division property of equality
B
factoring the binomial
C
completing the square
D
subtraction property of equality
4

What are the values of a, b, and c in the quadratic equation 0 = x2 – 3x – 2?

Question illustration
A
a = , b = 3, c = 2
Option A
B
a = , b = –3, c = –2
Option B
C
a = , b = 3, c = –2
Option C
D
a = , b = –3, c = 2
Option D
5

What is the value of the discriminant for the quadratic equation –3 = –x2 + 2x?Discriminant = b2 – 4ac

A
–8
B
4
C
8
D
16
6

Determine the discriminant for the quadratic equation -3=x2+4x+1. Based on the discriminant value, how many real number solutions does the equation have?Discriminant = b2-4ac

A
0
B
1
C
2
D
12
7

Determine the discriminant for the quadratic equation 0=-2x2+3. Based on the discriminant value, how many real number solutions does the equation have?Discriminant = b2-4ac

A
0
B
1
C
2
D
24
8

Which statement is true about the quadratic equation 8x2 − 5x + 3 = 0?

A
The constant term is 8.
B
The constant term is −5.
C
The leading coefficient is 8.
D
The leading coefficient is −5.
9

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

A
On a coordinate plane, a parabola opens up. It goes through (negative 1, 4), has a vertex at (1, 0), and goes through (3, 4).
Option A
B
On a coordinate plane, a parabola opens up. It goes through (negative 3, 4), has a vertex at (negative 1, 0), and goes through (1, 4).
Option B
C
On a coordinate plane, a parabola opens up. It goes through (negative 2, 5), has a vertex at (0, 1), and goes through (2, 5).
Option C
D
On a coordinate plane, a parabola opens up. It goes through (negative 2, 3), has a vertex at (0, negative 1), and goes through (2, 3).
Option D
10

Some of the steps in the derivation of the quadratic formula are shown.Step 3: –c + = aStep 4a: –c + = aStep 4b: + = a

Question illustration
A
factoring a polynomial
B
multiplication property of equality
C
converting to a common denominator
D
addition property of equality
11

Which statement is true about the quadratic equation 8x2 − 5x + 3 = 0?

A
The constant term is 8.
B
The constant term is −5.
C
The leading coefficient is 8.
D
The leading coefficient is −5.
12

What are the values of a, b, and c in the quadratic equation –2x2 + 4x – 3 = 0?

A
a = 2, b = 4, c = 3
B
a = 2, b = 4, c = –3
C
a = –2, b = 4, c = 3
D
a = –2, b = 4, c = – 3
13

The first step for deriving the quadratic formula from the quadratic equation, 0 = ax2 + bx + c, is shown.Step 1: –c = ax2 + bx

A
subtraction property of equality
B
completing the square
C
factoring out the constant
D
zero property of multiplication
14

Some of the steps in the derivation of the quadratic formula are shown.Step 4: Step 5: Step 6: Step 7:

Question illustration
A
Negative values, like −4ac, do not have a square root.
B
The ± symbol prevents the square root from being evaluated.
C
The square root of terms separated by addition and subtraction cannot be calculated individually.
D
The entire term b2 − 4ac must be divided by 2a before its square root can be determined.
15

A student is deriving the quadratic formula. Her first two steps are shown.Step 1: –c = ax2 + bxStep 2:

Question illustration
A
division property of equality
B
factoring the binomial
C
completing the square
D
subtraction property of equality
16

What is the value of the discriminant for the quadratic equation 0 = x + 2 + x2?Discriminant = b2 – 4ac

A
–9
B
–7
C
7
D
9
17

What is the value of the discriminant for the quadratic equation 0 = 2x2 + x – 3?Discriminant = b2 – 4ac

A
–25
B
–23
C
25
D
26
18

What is the value of the discriminant for the quadratic equation 0 = x + 2 + x2?Discriminant = b2 – 4ac

A
–9
B
–7
C
7
D
9
19

Which shows the correct substitution of the values a, b, and c from the equation 1 = –2x + 3x2 + 1 into the quadratic formula?Quadratic formula:

Question illustration
A
x = StartFraction negative (negative 2) plus or minus StartRoot (negative 2) squared minus 4 (3)(0) EndRoot Over 2(3) EndFraction
Option A
B
x = StartFraction negative (negative 2) plus or minus StartRoot (negative 2) squared minus 4 (3)(2) EndRoot Over 2(3) EndFraction
Option B
C
x = StartFraction negative (negative 2) plus or minus StartRoot (negative 2) squared minus 4 (3)(1) EndRoot Over 2(3) EndFraction
Option C
D
x = StartFraction negative 3 plus or minus StartRoot 3 squared minus 4 (negative 2)(0) EndRoot Over 2(negative 2) EndFraction
Option D
20

Determine the discriminant for the quadratic equation -3=x2+4x+1. Based on the discriminant value, how many real number solutions does the equation have?Discriminant = b2-4ac

A
0
B
1
C
2
D
12

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