Question

For the quadratic equation (x−7)^2=11 , complete the table by testing each of the given values to determine whether it is a solution. Identify the two solutions to the quadratic equation. (1 point) The smaller of the two solutions is x=() . The larger of the two solutions is x=() .

Answer 1

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:

1. \(x - 7 = \sqrt{11}\)
2. \(x - 7 = -\sqrt{11}\)

Now we solve each one for \(x\):

1. Adding 7 to both sides of the first equation:
\[
x = 7 + \sqrt{11}
\]

2. 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 \(x = 7 + \sqrt{11}\).