Yes, there is a mistake in Ariel's steps. Let's go through her work:
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She starts with the equation \( x^2 + 4x - 12 = 7 \).
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To put it in standard form, you should subtract 7 from both sides to set the equation to 0: \[ x^2 + 4x - 12 - 7 = 0 \implies x^2 + 4x - 19 = 0 \]
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Ariel incorrectly factored the expression to \( (x + 5)(x - 1) = 7 \). This is incorrect because the left-hand side doesn't equal 7 when factored like that.
To factor or solve the quadratic equation correctly, it should be understood that the correct factoring or solving method should follow from this standard form directly, or through the quadratic formula since it does not factor easily into integers.
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The quadratic formula could be used here: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = 4, c = -19 \).
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Compute the discriminant: \[ b^2 - 4ac = 4^2 - 4(1)(-19) = 16 + 76 = 92 \]
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Now apply the quadratic formula: \[ x = \frac{-4 \pm \sqrt{92}}{2} = \frac{-4 \pm 2\sqrt{23}}{2} = -2 \pm \sqrt{23} \]
Therefore, Ariel's mistake was in her factorization step, which led her to the incorrect solutions. The correct roots are \( x = -2 + \sqrt{23} \) and \( x = -2 - \sqrt{23} \).