To determine if each value is a solution to the quadratic equation \((x-7)^2 = 11\), we can solve the equation first and then substitute the values provided into the equation.
Step 1: Solve the quadratic equation
Starting with the equation:
\[ (x - 7)^2 = 11 \]
Take the square root of both sides:
\[ x - 7 = \pm \sqrt{11} \]
Now, solve for \(x\):
- \(x - 7 = \sqrt{11} \Rightarrow x = 7 + \sqrt{11}\)
- \(x - 7 = -\sqrt{11} \Rightarrow x = 7 - \sqrt{11}\)
So, the two solutions to the quadratic equation \((x-7)^2 = 11\) are:
- \(x = 7 + \sqrt{11}\)
- \(x = 7 - \sqrt{11}\)
Step 2: Complete the table
Now we need to evaluate the given expressions using \(x = 7\), \(x = 11\), \(x = 11 - \sqrt{7}\), \(x = 7 + \sqrt{11}\), \(x = 11 + \sqrt{7}\).
Calculating each value:
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For \(x = 7\): \[ (7 - 7)^2 = 0 \neq 11 \quad \text{False Statement} \]
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For \(x = 11\): \[ (11 - 7)^2 = 4 = 11 \quad \text{False Statement} \]
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For \(x = 11 - \sqrt{7}\): \[ (11 - \sqrt{7} - 7)^2 = (4 - \sqrt{7})^2 = 16 - 8\sqrt{7} + 7 = 23 - 8\sqrt{7} \neq 11 \quad \text{False Statement} \]
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For \(x = 7 + \sqrt{11}\): \[ \left((7 + \sqrt{11}) - 7\right)^2 = (\sqrt{11})^2 = 11 \quad \text{True Statement} \]
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For \(x = 11 + \sqrt{7}\): \[ (11 + \sqrt{7} - 7)^2 = (4 + \sqrt{7})^2 = 16 + 8\sqrt{7} + 7 = 23 + 8\sqrt{7} \neq 11 \quad \text{False Statement} \]
Final Table:
| x | Substituted | Evaluate | True Statement? | |----------------|----------------------------|-------------------------------|-------------------------| | 7 | \((7 - 7)^2\) | \(0\) | False | | 11 | \((11 - 7)^2\) | \(4\) | False | | \(11 - \sqrt{7}\) | \((11 - \sqrt{7} - 7)^2\) | \(23 - 8\sqrt{7}\) | False | | \(7 + \sqrt{11}\) | \((7 + \sqrt{11} - 7)^2\) | \(11\) | True | | \(11 + \sqrt{7}\) | \((11 + \sqrt{7} - 7)^2\) | \(23 + 8\sqrt{7}\) | False |
Conclusion
The two solutions to \((x-7)^2 = 11\) are:
- \(x = 7 + \sqrt{11}\)
- \(x = 7 - \sqrt{11}\)
In the table, only \(x = 7 + \sqrt{11}\) yields a true statement indicating it is a solution to the quadratic equation.