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Solve the quadratic equation 5x2+75x=0 by factoring out the GCF.(1 point) Responses The solutions are x=−15 and x=−5. The solut...Question
Solve the quadratic equation 5x2+75x=0 by factoring out the GCF.(1 point)
Responses
The solutions are x=−15 and x=−5.
The solutions are x equals negative 15 and x equals negative 5 .
The solutions are x=0 and x=15.
The solutions are x equals 0 and x equals 15 .
The solutions are x=−15 and x=0.
The solutions are x equals negative 15 and x equals 0 .
The solutions are x=0 and x=5.
Responses
The solutions are x=−15 and x=−5.
The solutions are x equals negative 15 and x equals negative 5 .
The solutions are x=0 and x=15.
The solutions are x equals 0 and x equals 15 .
The solutions are x=−15 and x=0.
The solutions are x equals negative 15 and x equals 0 .
The solutions are x=0 and x=5.
Answers
Answer
Which of the following tables shows the correct steps to factor out the GCF and solve the quadratic equation 7x2=56x?(1 point)
Responses
Put the quadratic in standard form 7x2−56x=0
Factor out the GCF 7x(x−8)=0
Set each factor equal to zero 7x=0 and x−8=0
The solutions are x=0 and x=8
Put the quadratic in standard form 7 x squared minus 56 x equals 0 Factor out the GCF 7 x left parenthesis x minus 8 right parenthesis equals 0 Set each factor equal to zero 7 x equals 0 and x minus 8 equals 0 The solutions are x equals 0 and x equals 8
Put the quadratic in standard form 7x2−56x=0
Factor out the GCF 7x(x−8)=0
Set each factor equal to zero 7x=0 and x−8=0
The solutions are x=−7 and x=8
Put the quadratic in standard form 7 x squared minus 56 x equals 0 Factor out the GCF 7 x left parenthesis x minus 8 right parenthesis equals 0 Set each factor equal to zero 7 x equals 0 and x minus 8 equals 0 The solutions are x equals negative 7 and x equals 8
Put the quadratic in standard form x−8=0
Factor out the GCF No GCF
Set each factor equal to zero x−8=0
The solutions are x=8
Put the quadratic in standard form x minus 8 equals 0 Factor out the GCF No GCF Set each factor equal to zero x minus 8 equals 0 The solutions are x equals 8
Put the quadratic in standard form 7x2+56x=0
Factor out the GCF 7x(x+8)=0
Set each factor equal to zero 7x=0 and x+8=0
The solutions are x=−8 and x=0
Responses
Put the quadratic in standard form 7x2−56x=0
Factor out the GCF 7x(x−8)=0
Set each factor equal to zero 7x=0 and x−8=0
The solutions are x=0 and x=8
Put the quadratic in standard form 7 x squared minus 56 x equals 0 Factor out the GCF 7 x left parenthesis x minus 8 right parenthesis equals 0 Set each factor equal to zero 7 x equals 0 and x minus 8 equals 0 The solutions are x equals 0 and x equals 8
Put the quadratic in standard form 7x2−56x=0
Factor out the GCF 7x(x−8)=0
Set each factor equal to zero 7x=0 and x−8=0
The solutions are x=−7 and x=8
Put the quadratic in standard form 7 x squared minus 56 x equals 0 Factor out the GCF 7 x left parenthesis x minus 8 right parenthesis equals 0 Set each factor equal to zero 7 x equals 0 and x minus 8 equals 0 The solutions are x equals negative 7 and x equals 8
Put the quadratic in standard form x−8=0
Factor out the GCF No GCF
Set each factor equal to zero x−8=0
The solutions are x=8
Put the quadratic in standard form x minus 8 equals 0 Factor out the GCF No GCF Set each factor equal to zero x minus 8 equals 0 The solutions are x equals 8
Put the quadratic in standard form 7x2+56x=0
Factor out the GCF 7x(x+8)=0
Set each factor equal to zero 7x=0 and x+8=0
The solutions are x=−8 and x=0
Answer
As Isla prepares to set off fireworks, she wants to ensure the spectators are a safe distance away. When shot in the air, one firework will follow the pathh(x)=−x2+15x, where h(x) is the height of the firework, and x is the horizontal distance it travels in feet. To determine how far the firework will travel before it reaches the ground, Isla sets the height equal to zero, and solves the following quadratic equation: 0=−x2+15x. How far will the firework travel before it reaches the ground?(1 point)
Responses
30 feet
30 feet
−15 feet
negative 15 feet
15 feet
15 feet
7.5 feet
7.5 feet
Responses
30 feet
30 feet
−15 feet
negative 15 feet
15 feet
15 feet
7.5 feet
7.5 feet
Answer
Hyung-eun is solving the quadratic equation x2−4x−12=0 by grouping. Her work is shown below. At which step does Hyung-eun first make an error?
x2−4x−12=0
Step1: x2+(−6x)+2x−12=0
Step 2: (x2+(−6x))+(2x−12)=0
Step 3: x(x+(−6))+2(x−6)=0
x(x−6)+2(x−6)=0
Step 4: (x+6)(x−2)=0
Step 5: x+6=0 or x−2=0
x=−6 or x=2
(1 point)
Responses
Step 4
Step 4
Step 1
Step 1
Step 3
Step 3
Step 2
Step 2
x2−4x−12=0
Step1: x2+(−6x)+2x−12=0
Step 2: (x2+(−6x))+(2x−12)=0
Step 3: x(x+(−6))+2(x−6)=0
x(x−6)+2(x−6)=0
Step 4: (x+6)(x−2)=0
Step 5: x+6=0 or x−2=0
x=−6 or x=2
(1 point)
Responses
Step 4
Step 4
Step 1
Step 1
Step 3
Step 3
Step 2
Step 2
Answer
Which of the following quadratic equations can be solved by grouping?(1 point)
Responses
x2+10x+21=0
x squared plus 10 x plus 21 equals 0
x2+8x−22=0
x squared plus 8 x minus 22 equals 0
x2−12x+18=0
x squared minus 12 x plus 18 equals 0
x2−4x−8=0
Responses
x2+10x+21=0
x squared plus 10 x plus 21 equals 0
x2+8x−22=0
x squared plus 8 x minus 22 equals 0
x2−12x+18=0
x squared minus 12 x plus 18 equals 0
x2−4x−8=0
Answered by
GPT-4o mini
To solve the quadratic equation \(5x^2 + 75x = 0\) by factoring out the greatest common factor (GCF), we can start by factoring out \(5x\):
\[
5x(x + 15) = 0
\]
Now, we set each factor equal to zero:
1. \(5x = 0\)
2. \(x + 15 = 0\)
From the first equation, we get:
\[
x = 0
\]
From the second equation, we subtract 15 from both sides:
\[
x = -15
\]
So, the solutions to the equation are \(x = 0\) and \(x = -15\).
The correct response is:
**The solutions are \(x = 0\) and \(x = -15\).**
\[
5x(x + 15) = 0
\]
Now, we set each factor equal to zero:
1. \(5x = 0\)
2. \(x + 15 = 0\)
From the first equation, we get:
\[
x = 0
\]
From the second equation, we subtract 15 from both sides:
\[
x = -15
\]
So, the solutions to the equation are \(x = 0\) and \(x = -15\).
The correct response is:
**The solutions are \(x = 0\) and \(x = -15\).**
Answered by
GPT-4o mini
To solve the equation \(7x^2 = 56x\), we first need to rewrite it in standard form:
1. **Put the quadratic in standard form:**
\[
7x^2 - 56x = 0
\]
2. **Factor out the GCF:**
The GCF of \(7x^2\) and \(56x\) is \(7x\), so we factor it out:
\[
7x(x - 8) = 0
\]
3. **Set each factor equal to zero:**
From \(7x = 0\) and \(x - 8 = 0\), we solve:
\(7x = 0 \rightarrow x = 0\)
\(x - 8 = 0 \rightarrow x = 8\)
4. **The solutions are:**
\(x = 0\) and \(x = 8\).
Among the provided responses, the following table correctly follows the steps to factor out the GCF and solve the quadratic equation:
**Response:**
- **Put the quadratic in standard form:** \(7x^2 - 56x = 0\)
- **Factor out the GCF:** \(7x(x - 8) = 0\)
- **Set each factor equal to zero:** \(7x = 0\) and \(x - 8 = 0\)
- **The solutions are:** \(x = 0\) and \(x = 8\)
1. **Put the quadratic in standard form:**
\[
7x^2 - 56x = 0
\]
2. **Factor out the GCF:**
The GCF of \(7x^2\) and \(56x\) is \(7x\), so we factor it out:
\[
7x(x - 8) = 0
\]
3. **Set each factor equal to zero:**
From \(7x = 0\) and \(x - 8 = 0\), we solve:
\(7x = 0 \rightarrow x = 0\)
\(x - 8 = 0 \rightarrow x = 8\)
4. **The solutions are:**
\(x = 0\) and \(x = 8\).
Among the provided responses, the following table correctly follows the steps to factor out the GCF and solve the quadratic equation:
**Response:**
- **Put the quadratic in standard form:** \(7x^2 - 56x = 0\)
- **Factor out the GCF:** \(7x(x - 8) = 0\)
- **Set each factor equal to zero:** \(7x = 0\) and \(x - 8 = 0\)
- **The solutions are:** \(x = 0\) and \(x = 8\)
Answered by
GPT-4o mini
To determine how far the firework travels before it reaches the ground, we need to solve the quadratic equation given by the height function \(h(x) = -x^2 + 15x\).
We set the height equal to zero:
\[
0 = -x^2 + 15x
\]
To solve the equation, we can first factor out the common terms:
\[
0 = -x(x - 15)
\]
Now, we set each factor equal to zero:
1. \(-x = 0 \implies x = 0\)
2. \(x - 15 = 0 \implies x = 15\)
The solutions to the equation are \(x = 0\) and \(x = 15\). The \(x = 0\) solution corresponds to the moment the firework was launched, and the \(x = 15\) solution corresponds to the horizontal distance traveled when it reaches the ground.
Thus, the firework will travel **15 feet** before it reaches the ground.
The correct response is:
**15 feet**.
We set the height equal to zero:
\[
0 = -x^2 + 15x
\]
To solve the equation, we can first factor out the common terms:
\[
0 = -x(x - 15)
\]
Now, we set each factor equal to zero:
1. \(-x = 0 \implies x = 0\)
2. \(x - 15 = 0 \implies x = 15\)
The solutions to the equation are \(x = 0\) and \(x = 15\). The \(x = 0\) solution corresponds to the moment the firework was launched, and the \(x = 15\) solution corresponds to the horizontal distance traveled when it reaches the ground.
Thus, the firework will travel **15 feet** before it reaches the ground.
The correct response is:
**15 feet**.
Answered by
GPT-4o mini
Let's go through Hyung-eun's work step by step to identify where she first made an error while solving the quadratic equation \(x^2 - 4x - 12 = 0\) by grouping.
1. **Starting equation:**
\(x^2 - 4x - 12 = 0\)
2. **Step 1:**
\(x^2 + (-6x) + 2x - 12 = 0\)
- This step is correct because the original equation can be broken down into parts that sum to \(-4x\).
3. **Step 2:**
\((x^2 + (-6x)) + (2x - 12) = 0\)
- This step is correct as it groups the terms correctly.
4. **Step 3:**
\(x(x + (-6)) + 2(x - 6) = 0\)
- This step is where the first error occurs. The correct factorization from \(2x - 12\) should be \(2(x - 6)\) rather than \(2(x - 6)\) since the grouping used \(-6\).
5. **Continuing Step 3 incorrectly:**
\(x(x - 6) + 2(x - 6) = 0\)
- This should not follow from the previous grouping since the original equation was not correctly factored.
6. **Step 4:**
\((x + 6)(x - 2) = 0\)
- Hyung-eun makes an incorrect assumption here linking back to the wrong grouping from Step 3.
7. **Final Step 5:**
\(x + 6 = 0\) or \(x - 2 = 0\) results in \(x = -6\) or \(x = 2\) which is incorrect due to prior mistakes.
### Conclusion
The first error occurs in **Step 3**, where she should have factored correctly from the terms obtained. Therefore, the correct response is:
**Step 3**.
1. **Starting equation:**
\(x^2 - 4x - 12 = 0\)
2. **Step 1:**
\(x^2 + (-6x) + 2x - 12 = 0\)
- This step is correct because the original equation can be broken down into parts that sum to \(-4x\).
3. **Step 2:**
\((x^2 + (-6x)) + (2x - 12) = 0\)
- This step is correct as it groups the terms correctly.
4. **Step 3:**
\(x(x + (-6)) + 2(x - 6) = 0\)
- This step is where the first error occurs. The correct factorization from \(2x - 12\) should be \(2(x - 6)\) rather than \(2(x - 6)\) since the grouping used \(-6\).
5. **Continuing Step 3 incorrectly:**
\(x(x - 6) + 2(x - 6) = 0\)
- This should not follow from the previous grouping since the original equation was not correctly factored.
6. **Step 4:**
\((x + 6)(x - 2) = 0\)
- Hyung-eun makes an incorrect assumption here linking back to the wrong grouping from Step 3.
7. **Final Step 5:**
\(x + 6 = 0\) or \(x - 2 = 0\) results in \(x = -6\) or \(x = 2\) which is incorrect due to prior mistakes.
### Conclusion
The first error occurs in **Step 3**, where she should have factored correctly from the terms obtained. Therefore, the correct response is:
**Step 3**.
Answered by
GPT-4o mini
To determine which of the given quadratic equations can be solved by grouping, we need to check if we can express the middle term as a sum of two terms whose product is equal to the product of the leading coefficient (which is 1) and the constant term.
The equation can generally be written in the form \(ax^2 + bx + c = 0\).
Let's evaluate each equation one by one:
1. **\(x^2 + 10x + 21 = 0\)**:
- The factors of 21 that add up to 10 are 3 and 7.
- This can be factored as \((x + 3)(x + 7) = 0\).
- This quadratic can be solved by factoring, but not specifically by grouping.
2. **\(x^2 + 8x - 22 = 0\)**:
- The factors of \(-22\) that add up to \(8\) are \(11\) and \(-2\).
- This can be factored as \((x + 11)(x - 2) = 0\).
- Not suitable for grouping in a standard way.
3. **\(x^2 - 12x + 18 = 0\)**:
- The factors of \(18\) that add up to \(-12\) are \(-6\) and \(-6\).
- This can be factored as \((x - 6)(x - 6) = 0\).
- This does not require grouping.
4. **\(x^2 - 4x - 8 = 0\)**:
- The product of the leading coefficient (1) and \(-8\) is \(-8\).
- We need two numbers that multiply to \(-8\) and add up to \(-4\). Those numbers are \(-6\) and \(2\).
- We can write:
\[
x^2 - 6x + 2x - 8 = 0
\]
- Now we can group:
\[
(x^2 - 6x) + (2x - 8) = 0
\]
- Factoring gives:
\[
x(x - 6) + 2(x - 4) = 0
\]
**Conclusion**:
The most suitable equation that can be factored by grouping is:
**\(x^2 - 4x - 8 = 0\)**.
The equation can generally be written in the form \(ax^2 + bx + c = 0\).
Let's evaluate each equation one by one:
1. **\(x^2 + 10x + 21 = 0\)**:
- The factors of 21 that add up to 10 are 3 and 7.
- This can be factored as \((x + 3)(x + 7) = 0\).
- This quadratic can be solved by factoring, but not specifically by grouping.
2. **\(x^2 + 8x - 22 = 0\)**:
- The factors of \(-22\) that add up to \(8\) are \(11\) and \(-2\).
- This can be factored as \((x + 11)(x - 2) = 0\).
- Not suitable for grouping in a standard way.
3. **\(x^2 - 12x + 18 = 0\)**:
- The factors of \(18\) that add up to \(-12\) are \(-6\) and \(-6\).
- This can be factored as \((x - 6)(x - 6) = 0\).
- This does not require grouping.
4. **\(x^2 - 4x - 8 = 0\)**:
- The product of the leading coefficient (1) and \(-8\) is \(-8\).
- We need two numbers that multiply to \(-8\) and add up to \(-4\). Those numbers are \(-6\) and \(2\).
- We can write:
\[
x^2 - 6x + 2x - 8 = 0
\]
- Now we can group:
\[
(x^2 - 6x) + (2x - 8) = 0
\]
- Factoring gives:
\[
x(x - 6) + 2(x - 4) = 0
\]
**Conclusion**:
The most suitable equation that can be factored by grouping is:
**\(x^2 - 4x - 8 = 0\)**.
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