To solve the equation \((x - 7)^2 = 11\), we first need to find the values of \(x\) that satisfy this equation.
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Taking the square root of both sides: \[ (x - 7) = \pm \sqrt{11} \]
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Solving for \(x\): For the positive root: \[ x - 7 = \sqrt{11} \implies x = 7 + \sqrt{11} \] For the negative root: \[ x - 7 = -\sqrt{11} \implies x = 7 - \sqrt{11} \]
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Finding the approximate values of the solutions: The value of \(\sqrt{11}\) is approximately 3.32.
So the two solutions are: \[ x = 7 + \sqrt{11} \approx 7 + 3.32 \approx 10.32 \] \[ x = 7 - \sqrt{11} \approx 7 - 3.32 \approx 3.68 \]
Now we can fill out the table with the evaluations of \(x = 7\), \(x = 11\), \(x = 7 + \sqrt{11}\), and \(x = 7 - \sqrt{11}\):
Table Completion:
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Substitute \(x = 7\): \[ (7 - 7)^2 = 0 \quad (\text{not a solution}) \]
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Substitute \(x = 11\): \[ (11 - 7)^2 = 4 \quad (\text{not a solution}) \]
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Substitute \(x = 7 + \sqrt{11}\): \[ ((7 + \sqrt{11}) - 7)^2 = (\sqrt{11})^2 = 11 \quad (\text{solution}) \]
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Substitute \(x = 7 - \sqrt{11}\): \[ ((7 - \sqrt{11}) - 7)^2 = (-\sqrt{11})^2 = 11 \quad (\text{solution}) \]
Conclusion:
The two solutions to the equation \((x - 7)^2 = 11\) are:
- The smaller solution is \(x = 7 - \sqrt{11}\)
- The larger solution is \(x = 7 + \sqrt{11}\)
Final Answer:
The smaller of the two solutions is \(x = 7 - \sqrt{11}\). The larger of the two solutions is \(x = 7 + \sqrt{11}\).
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