To find the solutions for the quadratic equation \((x - 7)^2 = 11\), we’ll first solve the equation for \(x\).
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Start by taking the square root of both sides: \[ x - 7 = \pm \sqrt{11} \]
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Now, solve for \(x\):
- Adding 7 to both sides gives two separate solutions: \[ x = 7 + \sqrt{11} \] \[ x = 7 - \sqrt{11} \]
Next, we need to evaluate these values and fill out your table.
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Calculate \(7 + \sqrt{11}\): \(\sqrt{11} \approx 3.3166\) (rounded) Therefore, \[ 7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166 \]
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Calculate \(7 - \sqrt{11}\): \[ 7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834 \]
Now let's identify the two solutions:
- Smaller solution: \(x = 7 - \sqrt{11} \approx 3.6834\)
- Larger solution: \(x = 7 + \sqrt{11} \approx 10.3166\)
Finally, placing the results in the answer format:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).
So,
- The smaller of the two solutions is \(x \approx 3.68\).
- The larger of the two solutions is \(x \approx 10.32\).
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