To solve the equation \( \sqrt{b^2 - 10^2} = 15 \), we will first isolate the square root and then square both sides to eliminate it.
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Start with the equation: \[ \sqrt{b^2 - 10^2} = 15 \]
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Square both sides: \[ b^2 - 10^2 = 15^2 \] This simplifies to: \[ b^2 - 100 = 225 \]
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Add 100 to both sides: \[ b^2 = 225 + 100 \] \[ b^2 = 325 \]
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Take the square root of both sides: \[ b = \pm \sqrt{325} \]
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Simplify \( \sqrt{325} \): \[ \sqrt{325} = \sqrt{25 \times 13} = \sqrt{25} \times \sqrt{13} = 5\sqrt{13} \]
So, the solutions for \( b \) are: \[ b = 5\sqrt{13} \quad \text{or} \quad b = -5\sqrt{13} \]