Question
Ariel is working on her math test. She is trying to solve the equation x^2+4x−12=7
Step 1: Begin with equation. x^2+4x−12=7
Step 2: Write quadratic equation as the product of two factors (x+5)(x−1)=7
Step 3: Set each factor equal to 0. x+5=0 and x−1=0
Step 4: Solve each equation. x=−5 and x=1
on step 2, she should have subtracted 7 from both side before factoring so that the solution is equal to 0.
on step 4, she has her signs wrong, she should not have made them opposites
on step 3, she has her signs wrong. both signs should be positive
on step1, she should have added 4x and -12 together
Step 1: Begin with equation. x^2+4x−12=7
Step 2: Write quadratic equation as the product of two factors (x+5)(x−1)=7
Step 3: Set each factor equal to 0. x+5=0 and x−1=0
Step 4: Solve each equation. x=−5 and x=1
on step 2, she should have subtracted 7 from both side before factoring so that the solution is equal to 0.
on step 4, she has her signs wrong, she should not have made them opposites
on step 3, she has her signs wrong. both signs should be positive
on step1, she should have added 4x and -12 together
Answers
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Answered by
GPT-4o mini
It seems there are a few misunderstandings in Ariel's steps for solving the quadratic equation \( x^2 + 4x - 12 = 7 \). Let's go through the steps correctly:
### Step 1: Start with the equation.
Ariel correctly starts with:
\[ x^2 + 4x - 12 = 7 \]
### Step 2: Rearrange the equation.
To set the equation to \(0\), we need to subtract \(7\) from both sides:
\[ x^2 + 4x - 12 - 7 = 0 \]
This simplifies to:
\[ x^2 + 4x - 19 = 0 \]
### Step 3: Factor the quadratic expression (if possible).
Ariel's next step was to factor, but typically we would check if factoring is feasible or use the quadratic formula since \( x^2 + 4x - 19 \) does not factor neatly.
### Step 4: Use the quadratic formula to find the roots.
For the equation \( ax^2 + bx + c = 0 \) where \( a = 1, b = 4, c = -19 \):
- The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now substituting in our values:
- \( b^2 - 4ac = 16 + 76 = 92 \)
Calculating further, we get:
\[ x = \frac{-4 \pm \sqrt{92}}{2} \]
\[ x = \frac{-4 \pm 2\sqrt{23}}{2} \]
\[ x = -2 \pm \sqrt{23} \]
### Final Solutions
Thus, the solutions to the equation \( x^2 + 4x - 19 = 0 \) are:
\[ x = -2 + \sqrt{23} \quad \text{and} \quad x = -2 - \sqrt{23} \]
So, summarizing the corrections:
- Step 1: Correct.
- Step 2: Correctly bring all terms to one side to set the equation to \(0\).
- Step 3: The equation does not factor neatly, so we would use the quadratic formula instead of setting any factors equal to zero.
- Step 4: The roots will be calculated correctly using the quadratic formula rather than finding opposites of factors.
Ariel should follow this corrected process to accurately solve the quadratic equation.
### Step 1: Start with the equation.
Ariel correctly starts with:
\[ x^2 + 4x - 12 = 7 \]
### Step 2: Rearrange the equation.
To set the equation to \(0\), we need to subtract \(7\) from both sides:
\[ x^2 + 4x - 12 - 7 = 0 \]
This simplifies to:
\[ x^2 + 4x - 19 = 0 \]
### Step 3: Factor the quadratic expression (if possible).
Ariel's next step was to factor, but typically we would check if factoring is feasible or use the quadratic formula since \( x^2 + 4x - 19 \) does not factor neatly.
### Step 4: Use the quadratic formula to find the roots.
For the equation \( ax^2 + bx + c = 0 \) where \( a = 1, b = 4, c = -19 \):
- The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now substituting in our values:
- \( b^2 - 4ac = 16 + 76 = 92 \)
Calculating further, we get:
\[ x = \frac{-4 \pm \sqrt{92}}{2} \]
\[ x = \frac{-4 \pm 2\sqrt{23}}{2} \]
\[ x = -2 \pm \sqrt{23} \]
### Final Solutions
Thus, the solutions to the equation \( x^2 + 4x - 19 = 0 \) are:
\[ x = -2 + \sqrt{23} \quad \text{and} \quad x = -2 - \sqrt{23} \]
So, summarizing the corrections:
- Step 1: Correct.
- Step 2: Correctly bring all terms to one side to set the equation to \(0\).
- Step 3: The equation does not factor neatly, so we would use the quadratic formula instead of setting any factors equal to zero.
- Step 4: The roots will be calculated correctly using the quadratic formula rather than finding opposites of factors.
Ariel should follow this corrected process to accurately solve the quadratic equation.
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