To solve the quadratic equation \((x + 36)^2 - 38 = 251\), we can begin by isolating the squared term. Here are the steps to solve it:
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Add 38 to both sides of the equation: \[ (x + 36)^2 - 38 + 38 = 251 + 38 \] \[ (x + 36)^2 = 289 \]
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Take the square root of both sides: \[ x + 36 = \pm \sqrt{289} \] Since \(\sqrt{289} = 17\), we have: \[ x + 36 = 17 \quad \text{or} \quad x + 36 = -17 \]
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Solve for \(x\) in both cases:
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For the first case: \[ x + 36 = 17 \] \[ x = 17 - 36 \] \[ x = -19 \]
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For the second case: \[ x + 36 = -17 \] \[ x = -17 - 36 \] \[ x = -53 \]
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Final answers: The two solutions are: \[ x = -19 \quad \text{and} \quad x = -53 \]
Thus, the solutions to the equation are \(x = -19\) and \(x = -53\).
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