Question
Solve the following quadratic equation by factoring out the greatest common factor (GCF): 2x2−14x=0
.(1 point)
The smaller solution is x=
, and the larger solution is x=
.
Remaining Attempts : 3
.(1 point)
The smaller solution is x=
, and the larger solution is x=
.
Remaining Attempts : 3
Answers
Answer
Rewrite the following quadratic equation in standard form and then solve by factoring out the GCF: 6x2=22x
.(1 point)
The smaller solution is x=
, and the larger solution is x=
.(1 point)
The smaller solution is x=
, and the larger solution is x=
Answer
During a water balloon fight, Louisa uses a slingshot to fire a water balloon over the fortified wall of the opponent. When fired from the slingshot, the balloon will follow the path h(x)=−x2+12x
, where h(x)
is the height of the balloon in feet, and x is the horizontal distance it will travel. To determine how far the balloon will travel before it reaches the ground, Louisa sets the height equal to zero and solves the quadratic equation 0=−x2+12x
. How far will the balloon travel before it reaches the ground?(1 point)
The balloon will travel
feet before it reaches the ground.
, where h(x)
is the height of the balloon in feet, and x is the horizontal distance it will travel. To determine how far the balloon will travel before it reaches the ground, Louisa sets the height equal to zero and solves the quadratic equation 0=−x2+12x
. How far will the balloon travel before it reaches the ground?(1 point)
The balloon will travel
feet before it reaches the ground.
Answer
Karim is solving the quadratic equation x2+5x−24=0
by grouping. His work is shown below. Determine the first step at which Karim makes an error.
Step 1: x2+8x+(−3x)−24=0
Step 2: (x2+8x)+(−3x−24)=0
Step 3: x(x+8)+3(x+8)=0
Step 4: (x+8)(x+3)=0
Step 5: x+8=0
or x+3=0
x=−8
or x=−3
by grouping. His work is shown below. Determine the first step at which Karim makes an error.
Step 1: x2+8x+(−3x)−24=0
Step 2: (x2+8x)+(−3x−24)=0
Step 3: x(x+8)+3(x+8)=0
Step 4: (x+8)(x+3)=0
Step 5: x+8=0
or x+3=0
x=−8
or x=−3
Answer
Karim is solving the quadratic equation x^2+5x−24=0
by grouping. His work is shown below. Determine the first step at which Karim makes an error.
Step 1: x^2+8x+(−3x)−24=0
Step 2: (x^2+8x)+(−3x−24)=0
Step 3: x(x+8)+3(x+8)=0
Step 4: (x+8)(x+3)=0
Step 5: x+8=0
or x+3=0
x=−8
or x=−3
(1 point)
Karim's first mistake occurs at Step
.
by grouping. His work is shown below. Determine the first step at which Karim makes an error.
Step 1: x^2+8x+(−3x)−24=0
Step 2: (x^2+8x)+(−3x−24)=0
Step 3: x(x+8)+3(x+8)=0
Step 4: (x+8)(x+3)=0
Step 5: x+8=0
or x+3=0
x=−8
or x=−3
(1 point)
Karim's first mistake occurs at Step
.
Answer
NO ITS STEP 3
Answered by
GPT-4o mini
To solve the quadratic equation \(2x^2 - 14x = 0\) by factoring out the greatest common factor (GCF), follow these steps:
1. Identify the GCF of the terms \(2x^2\) and \(-14x\). The GCF is \(2x\).
2. Factor out the GCF from the equation:
\[
2x(x - 7) = 0
\]
3. Set each factor equal to zero:
\[
2x = 0 \quad \text{or} \quad x - 7 = 0
\]
4. Solve for \(x\):
- From \(2x = 0\):
\[
x = 0
\]
- From \(x - 7 = 0\):
\[
x = 7
\]
Now we can identify the solutions:
- The smaller solution is \(x = 0\).
- The larger solution is \(x = 7\).
So:
- The smaller solution is \(x = 0\).
- The larger solution is \(x = 7\).
1. Identify the GCF of the terms \(2x^2\) and \(-14x\). The GCF is \(2x\).
2. Factor out the GCF from the equation:
\[
2x(x - 7) = 0
\]
3. Set each factor equal to zero:
\[
2x = 0 \quad \text{or} \quad x - 7 = 0
\]
4. Solve for \(x\):
- From \(2x = 0\):
\[
x = 0
\]
- From \(x - 7 = 0\):
\[
x = 7
\]
Now we can identify the solutions:
- The smaller solution is \(x = 0\).
- The larger solution is \(x = 7\).
So:
- The smaller solution is \(x = 0\).
- The larger solution is \(x = 7\).
Answered by
GPT-4o mini
To rewrite the equation \(6x^2 = 22x\) in standard form, we'll move all terms to one side of the equation:
1. Subtract \(22x\) from both sides:
\[
6x^2 - 22x = 0
\]
Next, we can factor out the greatest common factor (GCF):
2. Identify the GCF of the terms \(6x^2\) and \(-22x\). The GCF is \(2x\).
3. Factor out the GCF:
\[
2x(3x - 11) = 0
\]
4. Set each factor equal to zero:
\[
2x = 0 \quad \text{or} \quad 3x - 11 = 0
\]
5. Solve for \(x\):
- From \(2x = 0\):
\[
x = 0
\]
- From \(3x - 11 = 0\):
\[
3x = 11 \implies x = \frac{11}{3} \approx 3.67
\]
Now we can identify the solutions:
- The smaller solution is \(x = 0\).
- The larger solution is \(x = \frac{11}{3}\).
So:
- The smaller solution is \(x = 0\).
- The larger solution is \(x = \frac{11}{3}\).
1. Subtract \(22x\) from both sides:
\[
6x^2 - 22x = 0
\]
Next, we can factor out the greatest common factor (GCF):
2. Identify the GCF of the terms \(6x^2\) and \(-22x\). The GCF is \(2x\).
3. Factor out the GCF:
\[
2x(3x - 11) = 0
\]
4. Set each factor equal to zero:
\[
2x = 0 \quad \text{or} \quad 3x - 11 = 0
\]
5. Solve for \(x\):
- From \(2x = 0\):
\[
x = 0
\]
- From \(3x - 11 = 0\):
\[
3x = 11 \implies x = \frac{11}{3} \approx 3.67
\]
Now we can identify the solutions:
- The smaller solution is \(x = 0\).
- The larger solution is \(x = \frac{11}{3}\).
So:
- The smaller solution is \(x = 0\).
- The larger solution is \(x = \frac{11}{3}\).
Answered by
GPT-4o mini
To determine how far the balloon will travel before it reaches the ground, we need to solve the quadratic equation given by setting the height \(h(x)\) to zero:
\[
0 = -x^2 + 12x
\]
First, we can factor out the greatest common factor (GCF) from the equation:
1. The GCF is \( -x \):
\[
-x(x - 12) = 0
\]
Now, we set each factor equal to zero:
2. From \( -x = 0\):
\[
x = 0
\]
3. From \(x - 12 = 0\):
\[
x = 12
\]
Thus, the solutions to the equation are \(x = 0\) and \(x = 12\).
The balloon will reach the ground again at \(x = 12\) feet (after being fired from the slingshot at \(x = 0\)).
Therefore, the balloon will travel **12 feet** before it reaches the ground.
\[
0 = -x^2 + 12x
\]
First, we can factor out the greatest common factor (GCF) from the equation:
1. The GCF is \( -x \):
\[
-x(x - 12) = 0
\]
Now, we set each factor equal to zero:
2. From \( -x = 0\):
\[
x = 0
\]
3. From \(x - 12 = 0\):
\[
x = 12
\]
Thus, the solutions to the equation are \(x = 0\) and \(x = 12\).
The balloon will reach the ground again at \(x = 12\) feet (after being fired from the slingshot at \(x = 0\)).
Therefore, the balloon will travel **12 feet** before it reaches the ground.
Answered by
GPT-4o mini
Let's analyze each step of Karim's work to identify where he makes an error in solving the quadratic equation \(x^2 + 5x - 24 = 0\) by grouping.
1. **Step 1:** \(x^2 + 8x + (-3x) - 24 = 0\)
Karim is trying to rewrite the equation by splitting the middle term \(5x\). However, splitting \(5x\) as \(8x - 3x\) is incorrect because \(8 - 3\) does not equal \(5\). The correct split should be into terms that sum to \(5x\).
**Error:** The split should allow for terms that total \(5x\). For example, he could have used \(6x - x\), as \(6 + (-1) = 5\).
2. **Step 2:** \((x^2 + 8x) + (-3x - 24) = 0\)
This follows from Step 1 and retains the error of incorrectly splitting the middle term.
3. **Step 3:** \(x(x + 8) + 3(x + 8) = 0\)
Here, the factors of \(x^2 + 8x\) and \(-3x - 24\) should not yield \(x + 8\) because those should correspond to correct coefficients based on the terms in the original quadratic.
4. **Step 4:** \((x + 8)(x + 3) = 0\)
This is based on the incorrect grouping, thus producing incorrect factorization.
5. **Step 5:** \(x + 8 = 0\) or \(x + 3 = 0\)
The results from this erroneous step yield \(x = -8\) or \(x = -3\).
### Conclusion
**The first step at which Karim makes an error** is **Step 1**, where he incorrectly splits \(5x\) into \(8x - 3x\). The correct split should be such that the two terms can combine to give \(5x\). For example, he could have used \(6x - x\).
1. **Step 1:** \(x^2 + 8x + (-3x) - 24 = 0\)
Karim is trying to rewrite the equation by splitting the middle term \(5x\). However, splitting \(5x\) as \(8x - 3x\) is incorrect because \(8 - 3\) does not equal \(5\). The correct split should be into terms that sum to \(5x\).
**Error:** The split should allow for terms that total \(5x\). For example, he could have used \(6x - x\), as \(6 + (-1) = 5\).
2. **Step 2:** \((x^2 + 8x) + (-3x - 24) = 0\)
This follows from Step 1 and retains the error of incorrectly splitting the middle term.
3. **Step 3:** \(x(x + 8) + 3(x + 8) = 0\)
Here, the factors of \(x^2 + 8x\) and \(-3x - 24\) should not yield \(x + 8\) because those should correspond to correct coefficients based on the terms in the original quadratic.
4. **Step 4:** \((x + 8)(x + 3) = 0\)
This is based on the incorrect grouping, thus producing incorrect factorization.
5. **Step 5:** \(x + 8 = 0\) or \(x + 3 = 0\)
The results from this erroneous step yield \(x = -8\) or \(x = -3\).
### Conclusion
**The first step at which Karim makes an error** is **Step 1**, where he incorrectly splits \(5x\) into \(8x - 3x\). The correct split should be such that the two terms can combine to give \(5x\). For example, he could have used \(6x - x\).
Answered by
GPT-4o mini
Karim's first mistake occurs at **Step 1**.
In Step 1, he incorrectly splits the middle term \(5x\) into \(8x + (-3x)\). This is incorrect because \(8 - 3\) does not equal \(5\).
The correct way to split \(5x\) would be to use numbers that add up to \(5\) and multiply to give \(-24\) (the product of the coefficient of \(x^2\) and the constant term). For instance, \(8\) and \(-3\) do not satisfy the requirements as they will not correctly factor back to the original quadratic equation.
In summary, **the first mistake occurs at Step 1.**
In Step 1, he incorrectly splits the middle term \(5x\) into \(8x + (-3x)\). This is incorrect because \(8 - 3\) does not equal \(5\).
The correct way to split \(5x\) would be to use numbers that add up to \(5\) and multiply to give \(-24\) (the product of the coefficient of \(x^2\) and the constant term). For instance, \(8\) and \(-3\) do not satisfy the requirements as they will not correctly factor back to the original quadratic equation.
In summary, **the first mistake occurs at Step 1.**
Answered by
GPT-4o mini
I apologize for the confusion earlier. Let's go through Karim's work step by step again and focus on **Step 3**.
1. **Step 1:** \( x^2 + 8x + (-3x) - 24 = 0 \)
- Here, he incorrectly splits \(5x\) into \(8x + (-3x)\). Although this is a mistake, let's continue to the next step.
2. **Step 2:** \( (x^2 + 8x) + (-3x - 24) = 0 \)
- This step simply groups the terms but still retains the incorrect split.
3. **Step 3:** \( x(x + 8) + 3(x + 8) = 0 \)
- In this step, he has factored out the common terms; however, because the split in Step 1 was incorrect, the expression \( -3x - 24\) should factor to \( -3(x + 8)\) instead of \(3(x + 8)\). This incorrect group causes him to factor incorrectly.
4. **Step 4:** \( (x + 8)(x + 3) = 0 \)
- This is based on the failed factorization in Step 3.
5. **Step 5:** \( x + 8 = 0 \quad \text{or} \quad x + 3 = 0 \)
- Again, using incorrect factors from Step 4 leads to the solutions \(x = -8\) and \(x = -3\), which are also wrong.
So, indeed, the error in Karim's approach happens at **Step 3**, where he factors incorrectly due to the initial wrong split in Step 1.
Thank you for your patience, and I appreciate your clarification!
1. **Step 1:** \( x^2 + 8x + (-3x) - 24 = 0 \)
- Here, he incorrectly splits \(5x\) into \(8x + (-3x)\). Although this is a mistake, let's continue to the next step.
2. **Step 2:** \( (x^2 + 8x) + (-3x - 24) = 0 \)
- This step simply groups the terms but still retains the incorrect split.
3. **Step 3:** \( x(x + 8) + 3(x + 8) = 0 \)
- In this step, he has factored out the common terms; however, because the split in Step 1 was incorrect, the expression \( -3x - 24\) should factor to \( -3(x + 8)\) instead of \(3(x + 8)\). This incorrect group causes him to factor incorrectly.
4. **Step 4:** \( (x + 8)(x + 3) = 0 \)
- This is based on the failed factorization in Step 3.
5. **Step 5:** \( x + 8 = 0 \quad \text{or} \quad x + 3 = 0 \)
- Again, using incorrect factors from Step 4 leads to the solutions \(x = -8\) and \(x = -3\), which are also wrong.
So, indeed, the error in Karim's approach happens at **Step 3**, where he factors incorrectly due to the initial wrong split in Step 1.
Thank you for your patience, and I appreciate your clarification!
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