To solve the quadratic equation \((x - 7)^2 = 11\), we start by taking the square root of both sides:
\[ x - 7 = \pm \sqrt{11} \]
This gives us two equations:
- \(x - 7 = \sqrt{11}\)
- \(x - 7 = -\sqrt{11}\)
Now we solve each one for \(x\):
-
Adding 7 to both sides of the first equation: \[ x = 7 + \sqrt{11} \]
-
Adding 7 to both sides of the second equation: \[ x = 7 - \sqrt{11} \]
Now we have the two solutions:
- The larger of the two solutions is \(x = 7 + \sqrt{11}\)
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\)
Numerically, we can approximate \(\sqrt{11} \approx 3.3166\). Therefore:
- \(7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166\)
- \(7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834\)
So, the two solutions can be summarized as:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).
Final Answers:
The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
The larger of the two solutions is \(x = 7 + \sqrt{11}\).