If $\sqrt{x^2+{\sqrt{x^2+\sqrt{x^2+\sqrt{x^2+...}}}}}=9$ , where the given expression extends to an infinite number of roots, then what is the product of all the possible values of $x$ ?

Correct answer marked below

-72

Correct

3√2

6√2

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Explanation

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Official Explanation

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• Let

\sqrt{x^2 +\sqrt{x^2 +\sqrt{x^2 + \sqrt{x^2 + ...}}}} =9

…(i)

• Squaring both sides, we get:

x^2 + \sqrt{x^2 + \sqrt{x^2 + \sqrt{x^2 + \sqrt{x^2+...}}}} =81

• Substituting the value from equation (i):…

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What this question tests

Your handling of square and higher roots, including simplifying radicals.

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.