To solve the quadratic equation \((x + 36)^2 - 38 = 251\) using square roots, follow these steps:
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Isolate the squared term: \[ (x + 36)^2 - 38 = 251 \] Add 38 to both sides: \[ (x + 36)^2 = 251 + 38 \] \[ (x + 36)^2 = 289 \]
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Take the square root of both sides: \[ x + 36 = \pm \sqrt{289} \] Since \( \sqrt{289} = 17 \): \[ x + 36 = \pm 17 \]
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Solve for \(x\):
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For the positive case: \[ x + 36 = 17 \quad \Rightarrow \quad x = 17 - 36 \] \[ x = -19 \]
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For the negative case: \[ x + 36 = -17 \quad \Rightarrow \quad x = -17 - 36 \] \[ x = -53 \]
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So the solutions to the equation are: \[ { -19, -53 } \]
Thus, the answer is: \[ {−19,−53} \]