Asked by Yes.
Solve the factored quadratic equation (19x−3)(x−9)=0
(
19
𝑥
−
3
)
(
𝑥
−
9
)
=
0
.(1 point)
Responses
The solution set is x={−9, −319}
𝑥
=
{
−
9
,
−
3
19
}
.
The solution set is x={−9, −319}
𝑥
=
{
−
9
,
−
3
19
}
.
The solution set is x={3, 9}
𝑥
=
{
3
,
9
}
.
The solution set is x={3, 9}
𝑥
=
{
3
,
9
}
.
The solution set is x={319, 9}
𝑥
=
{
3
19
,
9
}
.
The solution set is x={319, 9}
𝑥
=
{
3
19
,
9
}
.
The solution set is x={193, 9}
𝑥
=
{
19
3
,
9
}
.
(
19
𝑥
−
3
)
(
𝑥
−
9
)
=
0
.(1 point)
Responses
The solution set is x={−9, −319}
𝑥
=
{
−
9
,
−
3
19
}
.
The solution set is x={−9, −319}
𝑥
=
{
−
9
,
−
3
19
}
.
The solution set is x={3, 9}
𝑥
=
{
3
,
9
}
.
The solution set is x={3, 9}
𝑥
=
{
3
,
9
}
.
The solution set is x={319, 9}
𝑥
=
{
3
19
,
9
}
.
The solution set is x={319, 9}
𝑥
=
{
3
19
,
9
}
.
The solution set is x={193, 9}
𝑥
=
{
19
3
,
9
}
.
Answers
Answered by
Yes.
Solve the quadratic equation 84x2+16x=0
by factoring out the GCF.(1 point)
Responses
The solutions are x=−421
The solutions are x=0 and x=421
The solutions are x=−4 and x=−421
The solutions are x=0 and x=4
by factoring out the GCF.(1 point)
Responses
The solutions are x=−421
The solutions are x=0 and x=421
The solutions are x=−4 and x=−421
The solutions are x=0 and x=4
Answered by
Yes.
Find the mistake made when solving x2−96=4x
x2−96=4x
Step 1: x2+4x−96=0
Add 4x to both sides.
Step 2: (x+12)(x−8)=0
Factor.
Step 3: x+12=0 or x−8=0
Use the Zero Product Property.
x=−12 or x=8
Responses:
Step 2 - She did not factor correctly.
Step 1 - She added 4x to both sides instead of subtracting.
Step 1 - She added 4 x to both sides instead of subtracting.
She did not make any mistakes.
Step 1 - She did not set it =0, so she did not use the Zero Product Property correctly.
x2−96=4x
Step 1: x2+4x−96=0
Add 4x to both sides.
Step 2: (x+12)(x−8)=0
Factor.
Step 3: x+12=0 or x−8=0
Use the Zero Product Property.
x=−12 or x=8
Responses:
Step 2 - She did not factor correctly.
Step 1 - She added 4x to both sides instead of subtracting.
Step 1 - She added 4 x to both sides instead of subtracting.
She did not make any mistakes.
Step 1 - She did not set it =0, so she did not use the Zero Product Property correctly.
Answered by
Yes.
Solve the following quadratic equation using the perfect square trinomial pattern:
x2+x+0.25=0 0.25
Responses
x=0.25
x=0.5
x=−0.25
x=−0.5
x2+x+0.25=0 0.25
Responses
x=0.25
x=0.5
x=−0.25
x=−0.5
Answered by
Yes.
Ginger was asked to solve 25x2−196=0 by factoring using the difference of squares pattern. What is/are the correct solution(s)?
Responses
x=196/25
x=14/5
x=−14/5, x=14/5
Responses
x=196/25
x=14/5
x=−14/5, x=14/5
Answered by
Yes.
Complete the square to solve the following quadratic equation: x2−6x−20=20
Responses
x=10, x=−4
x=−10, x=4
x=52, x=−46
Responses
x=10, x=−4
x=−10, x=4
x=52, x=−46
Answered by
GPT-4o mini
To solve the factored quadratic equation \((19x−3)(x−9)=0\), we can set each factor equal to zero.
1. Set the first factor to zero:
\[
19x - 3 = 0
\]
\[
19x = 3
\]
\[
x = \frac{3}{19}
\]
2. Set the second factor to zero:
\[
x - 9 = 0
\]
\[
x = 9
\]
So, the solutions to the equation are:
\[
x = \frac{3}{19} \quad \text{and} \quad x = 9
\]
Thus, the solution set is:
\[
x = \left\{ \frac{3}{19}, 9 \right\}
\]
Based on the response options provided, the correct solution set is:
**The solution set is \( x = \left\{ \frac{3}{19}, 9 \right\} \)** (though there may not be a perfectly matching option).
1. Set the first factor to zero:
\[
19x - 3 = 0
\]
\[
19x = 3
\]
\[
x = \frac{3}{19}
\]
2. Set the second factor to zero:
\[
x - 9 = 0
\]
\[
x = 9
\]
So, the solutions to the equation are:
\[
x = \frac{3}{19} \quad \text{and} \quad x = 9
\]
Thus, the solution set is:
\[
x = \left\{ \frac{3}{19}, 9 \right\}
\]
Based on the response options provided, the correct solution set is:
**The solution set is \( x = \left\{ \frac{3}{19}, 9 \right\} \)** (though there may not be a perfectly matching option).
Answered by
GPT-4o mini
To solve the quadratic equation \(84x^2 + 16x = 0\) by factoring out the greatest common factor (GCF), we first identify the GCF of the terms.
The GCF of \(84x^2\) and \(16x\) is \(4x\).
Now, we can factor the equation:
\[
84x^2 + 16x = 4x(21x + 4) = 0
\]
Next, we'll set each factor equal to zero:
1. From the factor \(4x = 0\):
\[
x = 0
\]
2. From the factor \(21x + 4 = 0\):
\[
21x = -4
\]
\[
x = -\frac{4}{21}
\]
Thus, the solutions to the equation \(84x^2 + 16x = 0\) are:
\[
x = 0 \quad \text{and} \quad x = -\frac{4}{21}
\]
Given the response options provided, the correct solution set is:
**The solutions are \(x = 0\) and \(x = -\frac{4}{21}\)** (though this option doesn't appear exactly in your list). If \( -\frac{4}{21} \) is meant to resemble the format for one of the other options, you might clarify or check them again, as they do not match the calculated solutions.
The GCF of \(84x^2\) and \(16x\) is \(4x\).
Now, we can factor the equation:
\[
84x^2 + 16x = 4x(21x + 4) = 0
\]
Next, we'll set each factor equal to zero:
1. From the factor \(4x = 0\):
\[
x = 0
\]
2. From the factor \(21x + 4 = 0\):
\[
21x = -4
\]
\[
x = -\frac{4}{21}
\]
Thus, the solutions to the equation \(84x^2 + 16x = 0\) are:
\[
x = 0 \quad \text{and} \quad x = -\frac{4}{21}
\]
Given the response options provided, the correct solution set is:
**The solutions are \(x = 0\) and \(x = -\frac{4}{21}\)** (though this option doesn't appear exactly in your list). If \( -\frac{4}{21} \) is meant to resemble the format for one of the other options, you might clarify or check them again, as they do not match the calculated solutions.
Answered by
GPT-4o mini
Let's analyze the provided steps for solving the equation \( x^2 - 96 = 4x \).
### Step 1: Rearranging the Equation
The original equation is:
\[
x^2 - 96 = 4x
\]
The first step involves moving all terms to one side:
\[
x^2 - 4x - 96 = 0
\]
In this step, instead of adding \(4x\) to both sides, it should have been subtracted from both sides to set the equation to zero correctly.
### Step 2: Factoring
The equation \(x^2 - 4x - 96 = 0\) can be factored as:
\[
(x + 12)(x - 8) = 0
\]
This step is correctly factored.
### Step 3: Zero Product Property
Using the Zero Product Property correctly gives two potential solutions:
\[
x + 12 = 0 \quad \Rightarrow \quad x = -12
\]
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
### Conclusion
The mistake in the solution occurs in **Step 1**, where it was incorrectly stated that \(4x\) was added to both sides instead of subtracted to bring it to the left side. Thus, the correct response is:
**Step 1 - She added \(4x\) to both sides instead of subtracting.**
### Step 1: Rearranging the Equation
The original equation is:
\[
x^2 - 96 = 4x
\]
The first step involves moving all terms to one side:
\[
x^2 - 4x - 96 = 0
\]
In this step, instead of adding \(4x\) to both sides, it should have been subtracted from both sides to set the equation to zero correctly.
### Step 2: Factoring
The equation \(x^2 - 4x - 96 = 0\) can be factored as:
\[
(x + 12)(x - 8) = 0
\]
This step is correctly factored.
### Step 3: Zero Product Property
Using the Zero Product Property correctly gives two potential solutions:
\[
x + 12 = 0 \quad \Rightarrow \quad x = -12
\]
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
### Conclusion
The mistake in the solution occurs in **Step 1**, where it was incorrectly stated that \(4x\) was added to both sides instead of subtracted to bring it to the left side. Thus, the correct response is:
**Step 1 - She added \(4x\) to both sides instead of subtracting.**
Answered by
GPT-4o mini
To solve the quadratic equation \(x^2 + x + 0.25 = 0\) using the perfect square trinomial pattern, we'll follow these steps:
1. **Recognize the perfect square trinomial:** The equation can be rewritten as:
\[
x^2 + x + \left(\frac{1}{2}\right)^2 = 0
\]
Here, \(\left(\frac{1}{2}\right)^2 = 0.25\).
2. **Rewrite the equation as a perfect square:**
\[
(x + 0.5)^2 = 0
\]
3. **Take the square root of both sides:**
\[
x + 0.5 = 0
\]
4. **Solve for \(x\):**
\[
x = -0.5
\]
Thus, the solution to the equation \(x^2 + x + 0.25 = 0\) is:
\[
\boxed{x = -0.5}
\]
Based on the provided responses, the correct solution is:
**\(x = -0.5\)**.
1. **Recognize the perfect square trinomial:** The equation can be rewritten as:
\[
x^2 + x + \left(\frac{1}{2}\right)^2 = 0
\]
Here, \(\left(\frac{1}{2}\right)^2 = 0.25\).
2. **Rewrite the equation as a perfect square:**
\[
(x + 0.5)^2 = 0
\]
3. **Take the square root of both sides:**
\[
x + 0.5 = 0
\]
4. **Solve for \(x\):**
\[
x = -0.5
\]
Thus, the solution to the equation \(x^2 + x + 0.25 = 0\) is:
\[
\boxed{x = -0.5}
\]
Based on the provided responses, the correct solution is:
**\(x = -0.5\)**.
Answered by
GPT-4o mini
To solve the equation \(25x^2 - 196 = 0\) by factoring using the difference of squares pattern, we can follow these steps:
1. **Recognize the difference of squares:**
The equation \(25x^2 - 196\) can be expressed as a difference of squares:
\[
(5x)^2 - (14)^2 = 0
\]
2. **Apply the difference of squares formula:**
The difference of squares can be factored as:
\[
(5x - 14)(5x + 14) = 0
\]
3. **Set each factor to zero:**
We set each factor equal to zero:
\[
5x - 14 = 0 \quad \text{or} \quad 5x + 14 = 0
\]
- For \(5x - 14 = 0\):
\[
5x = 14 \quad \Rightarrow \quad x = \frac{14}{5}
\]
- For \(5x + 14 = 0\):
\[
5x = -14 \quad \Rightarrow \quad x = -\frac{14}{5}
\]
4. **Write the solutions:**
The solutions are:
\[
x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5}
\]
Based on the response options provided, the correct solution is:
**\(x = -\frac{14}{5}, x = \frac{14}{5}\)**.
1. **Recognize the difference of squares:**
The equation \(25x^2 - 196\) can be expressed as a difference of squares:
\[
(5x)^2 - (14)^2 = 0
\]
2. **Apply the difference of squares formula:**
The difference of squares can be factored as:
\[
(5x - 14)(5x + 14) = 0
\]
3. **Set each factor to zero:**
We set each factor equal to zero:
\[
5x - 14 = 0 \quad \text{or} \quad 5x + 14 = 0
\]
- For \(5x - 14 = 0\):
\[
5x = 14 \quad \Rightarrow \quad x = \frac{14}{5}
\]
- For \(5x + 14 = 0\):
\[
5x = -14 \quad \Rightarrow \quad x = -\frac{14}{5}
\]
4. **Write the solutions:**
The solutions are:
\[
x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5}
\]
Based on the response options provided, the correct solution is:
**\(x = -\frac{14}{5}, x = \frac{14}{5}\)**.
Answered by
GPT-4o mini
To solve the quadratic equation \(x^2 - 6x - 20 = 20\) by completing the square, follow these steps:
1. **Simplify the equation:**
Start by moving \(20\) to the left side:
\[
x^2 - 6x - 20 - 20 = 0
\]
\[
x^2 - 6x - 40 = 0
\]
2. **Isolate the quadratic and linear terms:**
Add \(40\) to both sides:
\[
x^2 - 6x = 40
\]
3. **Complete the square:**
To complete the square, take half of the coefficient of \(x\) (which is \(-6\)), square it, and add it to both sides. Half of \(-6\) is \(-3\), and squaring it gives \(9\).
Add \(9\) to both sides:
\[
x^2 - 6x + 9 = 40 + 9
\]
\[
(x - 3)^2 = 49
\]
4. **Take the square root of both sides:**
\[
x - 3 = \pm 7
\]
5. **Solve for \(x\):**
- For \(x - 3 = 7\):
\[
x = 10
\]
- For \(x - 3 = -7\):
\[
x = -4
\]
Thus, the solutions to the equation \(x^2 - 6x - 20 = 20\) are:
\[
x = 10, \quad x = -4
\]
Based on the response options provided, the correct solution is:
**\(x = 10, x = -4\)**.
1. **Simplify the equation:**
Start by moving \(20\) to the left side:
\[
x^2 - 6x - 20 - 20 = 0
\]
\[
x^2 - 6x - 40 = 0
\]
2. **Isolate the quadratic and linear terms:**
Add \(40\) to both sides:
\[
x^2 - 6x = 40
\]
3. **Complete the square:**
To complete the square, take half of the coefficient of \(x\) (which is \(-6\)), square it, and add it to both sides. Half of \(-6\) is \(-3\), and squaring it gives \(9\).
Add \(9\) to both sides:
\[
x^2 - 6x + 9 = 40 + 9
\]
\[
(x - 3)^2 = 49
\]
4. **Take the square root of both sides:**
\[
x - 3 = \pm 7
\]
5. **Solve for \(x\):**
- For \(x - 3 = 7\):
\[
x = 10
\]
- For \(x - 3 = -7\):
\[
x = -4
\]
Thus, the solutions to the equation \(x^2 - 6x - 20 = 20\) are:
\[
x = 10, \quad x = -4
\]
Based on the response options provided, the correct solution is:
**\(x = 10, x = -4\)**.
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