To solve the quadratic equation \((x + 36)^2 - 38 = 251\), we will first isolate the squared term.
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Add 38 to both sides of the equation: \[ (x + 36)^2 = 251 + 38 \] \[ (x + 36)^2 = 289 \]
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Now, take the square root of both sides: \[ x + 36 = \pm \sqrt{289} \] \[ x + 36 = \pm 17 \]
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This gives us two cases to solve for \(x\):
Case 1: \[ x + 36 = 17 \] \[ x = 17 - 36 \] \[ x = -19 \]
Case 2: \[ x + 36 = -17 \] \[ x = -17 - 36 \] \[ x = -53 \]
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Therefore, the solutions to the equation are: \[ x = -19 \quad \text{and} \quad x = -53 \]
Final Answer: \(x = -19\) or \(x = -53\)