To solve the quadratic equation \((x - 7)^2 = 11\), we can take the square root of both sides and solve for \(x\).
Start by rearranging the equation: \[ (x - 7)^2 = 11 \] Taking the square root on both sides gives: \[ x - 7 = \pm \sqrt{11} \]
Now, solve for \(x\):
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For the positive case: \[ x - 7 = \sqrt{11} \implies x = 7 + \sqrt{11} \]
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For the negative case: \[ x - 7 = -\sqrt{11} \implies x = 7 - \sqrt{11} \]
Now we have the two potential solutions:
- The smaller solution is \(x = 7 - \sqrt{11}\).
- The larger solution is \(x = 7 + \sqrt{11}\).
Completing the Table
Now, let's evaluate the given expressions with respect to the two values we found:
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\(7 - \sqrt{11}\):
- Substituted: \(7 - \sqrt{11}\)
- Evaluate: \((7 - \sqrt{11} - 7)^2 = (-\sqrt{11})^2 = 11\)
- True statement? Yes, it satisfies the equation.
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\(7\):
- Substituted: \(7\)
- Evaluate: \((7 - 7)^2 = 0\), which is not equal to \(11\)
- True statement? No, it does not satisfy the equation.
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\(11 - 7\) (which simplifies to \(4\) but is irrelevant here):
- Substituted: \(4\)
- Evaluate: \((4 - 7)^2 = (-3)^2 = 9\), which is not equal to \(11\)
- True statement? No, it does not satisfy the equation.
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\(7 + \sqrt{11}\):
- Substituted: \(7 + \sqrt{11}\)
- Evaluate: \((7 + \sqrt{11} - 7)^2 = (\sqrt{11})^2 = 11\)
- True statement? Yes, it satisfies the equation.
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\(11 + 7\) (which simplifies to \(18\) but is irrelevant here):
- Substituted: \(18\)
- Evaluate: \((18 - 7)^2 = (11)^2 = 121\), which is not equal to \(11\)
- True statement? No, it does not satisfy the equation.
Conclusion
The results show that:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).
To summarize:
- The smaller solution is \(x = 7 - \sqrt{11}\).
- The larger solution is \(x = 7 + \sqrt{11}\).
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