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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

responses

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

1 answer

Let's analyze Ariel's steps in solving the equation \(x^2 + 4x - 12 = 7\).

  1. Step 1: Starting equation \(x^2 + 4x - 12 = 7\) is correct.

  2. Step 2: This step is incorrect. To factor the quadratic equation, Ariel should first rearrange the equation to set it equal to zero. This means subtracting 7 from both sides:

    \[ x^2 + 4x - 12 - 7 = 0 \implies x^2 + 4x - 19 = 0 \]

    Then it may be factored (or solved using the quadratic formula) as a next step. The expression \(x^2 + 4x - 19\) does not factor nicely, so she will ultimately need to apply the quadratic formula instead.

  3. Step 3: If we proceed to set the equation equal to zero correctly, there would not be factors of \(x + 5\) and \(x - 1\) since the correct equation is \(x^2 + 4x - 19 = 0\). If she were trying to set each factor equal to zero after incorrect factoring, this would not make sense.

  4. Step 4: Therefore, since the equation has not been factored correctly in Step 2, she cannot proceed to find the wrong roots \(x = -5\) and \(x = 1\).

Correct Steps to Solve the Equation:

  1. Start with: \[ x^2 + 4x - 12 = 7 \]

  2. Rearrange to: \[ x^2 + 4x - 19 = 0 \]

  3. Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 1\), \(b = 4\), and \(c = -19\).

  4. Calculate the discriminant: \[ b^2 - 4ac = 4^2 - 4(1)(-19) = 16 + 76 = 92 \]

  5. Now substitute into the formula: \[ x = \frac{-4 \pm \sqrt{92}}{2} = \frac{-4 \pm 2\sqrt{23}}{2} = -2 \pm \sqrt{23} \]

  6. Final answers: \[ x = -2 + \sqrt{23} \quad \text{or} \quad x = -2 - \sqrt{23} \]

Summary:

Ariel made several errors in her approach, especially in the factoring and setting equal to zero. Correcting these steps leads to the use of the quadratic formula for the correct solution.