Let's go through the calculations step by step for clarity.
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Start with your equation: \[ (\sqrt{1 + x^2})^2 = 5^2 \]
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Simplify both sides: \[ 1 + x^2 = 25 \]
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Subtract 1 from both sides: \[ x^2 = 25 - 1 \] \[ x^2 = 24 \]
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Take the square root of both sides: \[ x = \sqrt{24} \quad \text{or} \quad x = -\sqrt{24} \]
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Simplifying \(\sqrt{24}\): \[ \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} \]
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Therefore, the solutions are: \[ x = 2\sqrt{6} \quad \text{or} \quad x = -2\sqrt{6} \]
Your final answers are correct: \[ x = 2\sqrt{6} \quad \text{or} \quad x = -2\sqrt{6} \]