To solve the equation \(x^2 + 2x - 36 = 9x + 24\) by factoring, follow these steps:
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Rearrange the equation by moving all terms to one side: \[ x^2 + 2x - 36 - 9x - 24 = 0 \] Simplifying this gives: \[ x^2 + 2x - 9x - 36 - 24 = 0 \] Combine like terms: \[ x^2 - 7x - 60 = 0 \]
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Factor the quadratic: We need to factor \(x^2 - 7x - 60\). We are looking for two numbers that multiply to \(-60\) and add to \(-7\). The numbers \(-12\) and \(5\) meet these criteria.
Thus, we can factor the quadratic as: \[ (x - 12)(x + 5) = 0 \]
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Set each factor to zero: Now, solve for \(x\) by setting each factor equal to zero: \[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \] \[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \]
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Final solution: The solutions to the equation \(x^2 + 2x - 36 = 9x + 24\) are: \[ x = 12 \quad \text{and} \quad x = -5. \]