To determine which of the given quadratic equations can be solved by factoring, we need to bring each equation to the standard form \( ax^2 + bx + c = 0 \) and see if they can be factored easily.
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Equation: \( x^2 = 4x - 10 \)
Rearranging gives: \[ x^2 - 4x + 10 = 0 \] The discriminant (\( D \)) is \( b^2 - 4ac = (-4)^2 - 4(1)(10) = 16 - 40 = -24 \), which is negative. This means it cannot be factored over the real numbers.
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Equation: \( x^2 = 2x + 8 \)
Rearranging gives: \[ x^2 - 2x - 8 = 0 \] This can be factored as: \[ (x - 4)(x + 2) = 0 \] So this equation can be solved by factoring.
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Equation: \( x^2 + 9x = 8 \)
Rearranging gives: \[ x^2 + 9x - 8 = 0 \] We need to check if this can be factored. The discriminant is \( D = 9^2 - 4(1)(-8) = 81 + 32 = 113 \), which is not a perfect square, indicating it cannot be factored easily.
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Equation: \( -12 = x^2 - x \)
Rearranging gives: \[ x^2 - x + 12 = 0 \] The discriminant is \( D = (-1)^2 - 4(1)(12) = 1 - 48 = -47 \), which is negative. Therefore, this also cannot be factored over the real numbers.
Conclusion:
The only equation that can be solved by factoring is:
\( x^2 = 2x + 8 \).