To solve the equation \((x + 36)^2 - 38 = 251\), we will first isolate the squared term and then take the square root of both sides.
- Add 38 to both sides:
\[ (x + 36)^2 = 251 + 38 \] \[ (x + 36)^2 = 289 \]
- Take the square root of both sides (remembering to consider both the positive and negative roots):
\[ x + 36 = \pm \sqrt{289} \] \[ x + 36 = \pm 17 \]
- Solve for \(x\) in each case:
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First case: \[ x + 36 = 17 \] \[ x = 17 - 36 \] \[ x = -19 \]
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Second case: \[ x + 36 = -17 \] \[ x = -17 - 36 \] \[ x = -53 \]
- Combine the results: So the solutions are \(x = -19\) and \(x = -53\).
Thus, the solution set is \({-19, -53}\).
Since the answer is not available among the options provided, let's check which one might have a typographical error:
A: { 253, -325}
B: { 17.84 , - 13.84}
C: { -10 , -53}
D: { 53, 19}
Based on our derived values, none of the provided answer options match. It appears there may be an inconsistency with the options provided. However, the correct solutions derived from the quadratic equation are indeed:
\(-19\) and \(-53\).
If it had been intended to be in a list resembling the answer choices, the closest correct solution simply would be \( {-19, -53}\).
Please double-check the options or the original problem for any errors.