To solve the equation \((x-7)^2 = 11\), we can start by taking the square root of both sides. Remember to consider both the positive and negative square roots:
\[ x - 7 = \sqrt{11} \quad \text{or} \quad x - 7 = -\sqrt{11} \]
Now, we can solve for \(x\) in each case.
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For the first equation: \[ x - 7 = \sqrt{11} \] \[ x = \sqrt{11} + 7 \]
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For the second equation: \[ x - 7 = -\sqrt{11} \] \[ x = -\sqrt{11} + 7 \]
Thus, the solutions to the equation \((x-7)^2 = 11\) are:
\[ x = 7 + \sqrt{11} \quad \text{and} \quad x = 7 - \sqrt{11} \]