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.