Problem

(b) $\sqrt{x-6}=x-6$

Answer

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Answer

Final Answer: The solutions to the equation \(\sqrt{x-6}=x-6\) are \(\boxed{6}\) and \(\boxed{7}\)

Steps

Step 1 ：Given the equation \(\sqrt{x-6}=x-6\)

Step 2 ：Square both sides of the equation to eliminate the square root, resulting in \(x^2 - 2x\sqrt{x - 6} - 11x + 12\sqrt{x - 6} + 30\)

Step 3 ：Solve the resulting quadratic equation to find the potential solutions, which are \(x = 6\) and \(x = 7\)

Step 4 ：Check these solutions in the original equation to ensure they are not extraneous solutions. Both \(x = 6\) and \(x = 7\) are valid solutions

Step 5 ：Final Answer: The solutions to the equation \(\sqrt{x-6}=x-6\) are \(\boxed{6}\) and \(\boxed{7}\)

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