To solve the quadratic equation \((x - 6)^2 = 51\) using square roots, follow these steps:
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Take the square root of both sides: \[ x - 6 = \pm \sqrt{51} \]
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Calculate \(\sqrt{51}\): The approximate value of \(\sqrt{51}\) is about 7.14 (since \(7.14^2 \approx 51\)).
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Rewrite the equation as two separate equations: \[ x - 6 = \sqrt{51} \quad \text{and} \quad x - 6 = -\sqrt{51} \]
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Solve for \(x\) in both cases:
- For the first equation: \[ x = 6 + \sqrt{51} \]
- For the second equation: \[ x = 6 - \sqrt{51} \]
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Now substitute \(\sqrt{51}\) with its approximated value (7.14):
- First solution: \[ x = 6 + 7.14 \approx 13.14 \]
- Second solution: \[ x = 6 - 7.14 \approx -1.14 \]
Thus, the two solutions to the equation \((x - 6)^2 = 51\) are: \[ x \approx 13.14 \quad \text{and} \quad x \approx -1.14 \]
Rounded to the nearest hundredth, the solutions are: \[ \boxed{13.14} \quad \text{and} \quad \boxed{-1.14} \]