Factor each expression: \( x^{2}-8 x+15=\square \) \( x^{2}-3 x-28=\square \) \( x^{2}+5 x-14=\square \)
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Beyond the Answer
To factor the expressions, we can find two numbers that multiply to the constant term (the last number) and add up to the coefficient of the linear term (the middle number). 1. For \( x^{2}-8x+15 \), we look for numbers that multiply to \( 15 \) and add up to \( -8 \). The numbers \( -3 \) and \( -5 \) do the trick, so we have: \[ x^{2} - 8x + 15 = (x - 3)(x - 5) \] 2. For \( x^{2}-3x-28 \), we need numbers that multiply to \( -28 \) and add to \( -3 \). The numbers \( 4 \) and \( -7 \) work, yielding: \[ x^{2} - 3x - 28 = (x + 4)(x - 7) \] 3. Lastly, for \( x^{2} + 5x - 14 \), we want numbers that multiply to \( -14 \) and add to \( 5 \). The numbers \( 7 \) and \( -2 \) fit, resulting in: \[ x^{2} + 5x - 14 = (x + 7)(x - 2) \] So, we can summarize the factorizations as follows: - \( x^{2}-8x+15 = (x - 3)(x - 5) \) - \( x^{2}-3x-28 = (x + 4)(x - 7) \) - \( x^{2}+5x-14 = (x + 7)(x - 2) \)
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