Question
Determine whether the values 10 and 30 are solutions to the quadratic equation (x−27)(x−10)=60 .(1 point)
Responses
Only x=10 is a solution.
Only x equals 10 is a solution.
Only x=30 is a solution.
Only x equals 30 is a solution.
Both x=10 and x=30 are solutions.
Both x equals 10 and x equals 30 are solutions.
Neither x=10 nor x=30 is a solution.
Responses
Only x=10 is a solution.
Only x equals 10 is a solution.
Only x=30 is a solution.
Only x equals 30 is a solution.
Both x=10 and x=30 are solutions.
Both x equals 10 and x equals 30 are solutions.
Neither x=10 nor x=30 is a solution.
Answers
Answer
What type(s) of value(s) can a negative number produce when squared?(1 point)
Responses
zero only
zero only
positive values only
positive values only
positive or negative values
positive or negative values
negative values only
Responses
zero only
zero only
positive values only
positive values only
positive or negative values
positive or negative values
negative values only
Answer
Use inspection to solve the equation x2=196
, then select the correct solution set below. If there are no real solutions, select “There are no real solutions.”(1 point)
Responses
{−14,14}
left brace negative 14 comma 14 right brace
{−14}
left brace negative 14 right brace
There are no real solutions.
There are no real solutions.
{14}
, then select the correct solution set below. If there are no real solutions, select “There are no real solutions.”(1 point)
Responses
{−14,14}
left brace negative 14 comma 14 right brace
{−14}
left brace negative 14 right brace
There are no real solutions.
There are no real solutions.
{14}
Answer
Solve the following quadratic equation using square roots: (x+12)2=9
.(1 point)
Responses
{−3,−21}
left brace negative 3 comma negative 21 right brace
{−9,−15}
left brace negative 9 comma negative 15 right brace
{−7.5,−16.5}
left brace negative 7.5 comma negative 16.5 right brace
{15,9}
.(1 point)
Responses
{−3,−21}
left brace negative 3 comma negative 21 right brace
{−9,−15}
left brace negative 9 comma negative 15 right brace
{−7.5,−16.5}
left brace negative 7.5 comma negative 16.5 right brace
{15,9}
Answer
Based on the Zero Product Property, if either of the two factors in a quadratic equation equals zero, the entire equation equals zero. Find the values of x that make the following quadratic equation equal 0: (x−1)(2x+5)=0
.(1 point)
Responses
The quadratic equation will equal zero when x=−5
and x=1
.
The quadratic equation will equal zero when x equals negative 5 and x equals 1 .
The quadratic equation will equal zero when x=−1
and x=2/5
.
The quadratic equation will equal zero when x equals negative 1 and x equals Start Fraction 2 over 5 End Fraction .
The quadratic equation will equal zero when x=0
and x=1
.
The quadratic equation will equal zero when x equals 0 and x equals 1 .
The quadratic equation will equal zero when x=−5/2
and x=1
.
.(1 point)
Responses
The quadratic equation will equal zero when x=−5
and x=1
.
The quadratic equation will equal zero when x equals negative 5 and x equals 1 .
The quadratic equation will equal zero when x=−1
and x=2/5
.
The quadratic equation will equal zero when x equals negative 1 and x equals Start Fraction 2 over 5 End Fraction .
The quadratic equation will equal zero when x=0
and x=1
.
The quadratic equation will equal zero when x equals 0 and x equals 1 .
The quadratic equation will equal zero when x=−5/2
and x=1
.
Answer
Solve the factored quadratic equation (13x−2)(x−34)=0
.(1 point)
Responses
The solution set is x={−34,−2/13}
.
The solution set is x equals left brace negative 34 comma negative Start Fraction 2 over 13 End Fraction right brace .
The solution set is x={2/13,34}
.
The solution set is x equals left brace Start Fraction 2 over 13 End Fraction comma 34 right brace .
The solution set is x={13/2,34}
.
The solution set is x equals left brace Start Fraction 13 over 2 End Fraction comma 34 right brace .
The solution set is x={2,34}
.
.(1 point)
Responses
The solution set is x={−34,−2/13}
.
The solution set is x equals left brace negative 34 comma negative Start Fraction 2 over 13 End Fraction right brace .
The solution set is x={2/13,34}
.
The solution set is x equals left brace Start Fraction 2 over 13 End Fraction comma 34 right brace .
The solution set is x={13/2,34}
.
The solution set is x equals left brace Start Fraction 13 over 2 End Fraction comma 34 right brace .
The solution set is x={2,34}
.
Answered by
GPT-4o mini
To determine whether the values \( x = 10 \) and \( x = 30 \) are solutions to the quadratic equation \( (x − 27)(x − 10) = 60 \), we can substitute each value into the equation and check if the equation holds true.
1. **Check \( x = 10 \):**
\[
(10 - 27)(10 - 10) = 60
\]
\[
(-17)(0) = 60
\]
\[
0 \neq 60
\]
Thus, \( x = 10 \) is *not* a solution.
2. **Check \( x = 30 \):**
\[
(30 - 27)(30 - 10) = 60
\]
\[
(3)(20) = 60
\]
\[
60 = 60
\]
Thus, \( x = 30 \) *is* a solution.
Based on the results, we can conclude that:
**Only \( x = 30 \) is a solution.**
1. **Check \( x = 10 \):**
\[
(10 - 27)(10 - 10) = 60
\]
\[
(-17)(0) = 60
\]
\[
0 \neq 60
\]
Thus, \( x = 10 \) is *not* a solution.
2. **Check \( x = 30 \):**
\[
(30 - 27)(30 - 10) = 60
\]
\[
(3)(20) = 60
\]
\[
60 = 60
\]
Thus, \( x = 30 \) *is* a solution.
Based on the results, we can conclude that:
**Only \( x = 30 \) is a solution.**
Answered by
GPT-4o mini
When a negative number is squared, the result is always positive. This is because squaring a number involves multiplying it by itself, and since the product of two negative numbers is positive, the square of a negative number is also positive.
Therefore, the correct response is:
**positive values only**.
Therefore, the correct response is:
**positive values only**.
Answered by
GPT-4o mini
To solve the equation \( x^2 = 196 \) using inspection, we want to find the values of \( x \) such that when squared, they equal 196.
Taking the square root of both sides, we find:
\[
x = \pm \sqrt{196}
\]
Calculating the square root:
\[
\sqrt{196} = 14
\]
This gives us two possible solutions:
\[
x = 14 \quad \text{and} \quad x = -14
\]
Thus, the solution set is:
\(\{ -14, 14 \}\)
The correct answer is:
**{−14, 14}**.
Taking the square root of both sides, we find:
\[
x = \pm \sqrt{196}
\]
Calculating the square root:
\[
\sqrt{196} = 14
\]
This gives us two possible solutions:
\[
x = 14 \quad \text{and} \quad x = -14
\]
Thus, the solution set is:
\(\{ -14, 14 \}\)
The correct answer is:
**{−14, 14}**.
Answered by
GPT-4o mini
To solve the quadratic equation \((x + 12)^2 = 9\) using square roots, we can follow these steps:
1. Take the square root of both sides:
\[
x + 12 = \pm \sqrt{9}
\]
\[
x + 12 = \pm 3
\]
2. This gives us two equations to solve:
- \(x + 12 = 3\)
- \(x + 12 = -3\)
3. Solve each equation:
For \(x + 12 = 3\):
\[
x = 3 - 12
\]
\[
x = -9
\]
For \(x + 12 = -3\):
\[
x = -3 - 12
\]
\[
x = -15
\]
Thus, the solutions are \(x = -9\) and \(x = -15\).
The correct solution set is:
**{−9,−15}**.
1. Take the square root of both sides:
\[
x + 12 = \pm \sqrt{9}
\]
\[
x + 12 = \pm 3
\]
2. This gives us two equations to solve:
- \(x + 12 = 3\)
- \(x + 12 = -3\)
3. Solve each equation:
For \(x + 12 = 3\):
\[
x = 3 - 12
\]
\[
x = -9
\]
For \(x + 12 = -3\):
\[
x = -3 - 12
\]
\[
x = -15
\]
Thus, the solutions are \(x = -9\) and \(x = -15\).
The correct solution set is:
**{−9,−15}**.
Answered by
GPT-4o mini
To find the values of \( x \) that make the equation \( (x - 1)(2x + 5) = 0 \) equal zero, we can use the Zero Product Property. This property states that if the product of two factors is zero, then at least one of the factors must be equal to zero.
Set each factor equal to zero:
1. **For the first factor:**
\[
x - 1 = 0
\]
Solving for \( x \):
\[
x = 1
\]
2. **For the second factor:**
\[
2x + 5 = 0
\]
Solving for \( x \):
\[
2x = -5 \\
x = -\frac{5}{2}
\]
Thus, the values of \( x \) that make the equation equal zero are:
\[
x = 1 \quad \text{and} \quad x = -\frac{5}{2}
\]
The correct response is:
**The quadratic equation will equal zero when \( x = -\frac{5}{2} \) and \( x = 1 \).**
Set each factor equal to zero:
1. **For the first factor:**
\[
x - 1 = 0
\]
Solving for \( x \):
\[
x = 1
\]
2. **For the second factor:**
\[
2x + 5 = 0
\]
Solving for \( x \):
\[
2x = -5 \\
x = -\frac{5}{2}
\]
Thus, the values of \( x \) that make the equation equal zero are:
\[
x = 1 \quad \text{and} \quad x = -\frac{5}{2}
\]
The correct response is:
**The quadratic equation will equal zero when \( x = -\frac{5}{2} \) and \( x = 1 \).**
Answered by
GPT-4o mini
To solve the factored quadratic equation \( (13x - 2)(x - 34) = 0 \) using the Zero Product Property, we'll set each factor equal to zero:
1. **For the first factor:**
\[
13x - 2 = 0
\]
Solving for \( x \):
\[
13x = 2 \\
x = \frac{2}{13}
\]
2. **For the second factor:**
\[
x - 34 = 0
\]
Solving for \( x \):
\[
x = 34
\]
Thus, the solutions are:
\[
x = \frac{2}{13} \quad \text{and} \quad x = 34.
\]
The solution set is:
\[
x = \left\{ \frac{2}{13}, 34 \right\}.
\]
Therefore, the correct response is:
**The solution set is \( x = \left\{ \frac{2}{13}, 34 \right\} \).**
1. **For the first factor:**
\[
13x - 2 = 0
\]
Solving for \( x \):
\[
13x = 2 \\
x = \frac{2}{13}
\]
2. **For the second factor:**
\[
x - 34 = 0
\]
Solving for \( x \):
\[
x = 34
\]
Thus, the solutions are:
\[
x = \frac{2}{13} \quad \text{and} \quad x = 34.
\]
The solution set is:
\[
x = \left\{ \frac{2}{13}, 34 \right\}.
\]
Therefore, the correct response is:
**The solution set is \( x = \left\{ \frac{2}{13}, 34 \right\} \).**