Question

\begin{tabular}{|c|c|c|c|}
\hline\( x \) & Substituted & Evaluate & True Statement? \\
\hline \( 7-\sqrt{11} \) & & & \\
\hline 7 & & & \\
\hline \( 11-\sqrt{7} \) & & & \\
\hline \( 7+\sqrt{11} \) & & & \\
\hline \( 11+\sqrt{7} \) & & & \\
\hline
\end{tabular}
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 \( x=\square \). The larger of the two solutions is \( x=\square \).

Answer 1

To begin, we need to solve the quadratic equation \( (x-7)^2 = 11 \).

Taking the square root of both sides, we obtain:

\[
x - 7 = \pm \sqrt{11}
\]

From this, we can solve for \( x \):

1. \( x - 7 = \sqrt{11} \):
\[
x = 7 + \sqrt{11}
\]

2. \( x - 7 = -\sqrt{11} \):
\[
x = 7 - \sqrt{11}
\]

So the solutions to the quadratic equation are \( x = 7 + \sqrt{11} \) and \( x = 7 - \sqrt{11} \).

Now, let's evaluate each of the given values in the table.

### Evaluating Each Value

1. **For \( x = 7 - \sqrt{11} \)**:
- Substituting:
\[
(7 - \sqrt{11} - 7)^2 = (-\sqrt{11})^2 = 11
\]
- True statement? Yes.

2. **For \( x = 7 \)**:
- Substituting:
\[
(7 - 7)^2 = 0 \neq 11
\]
- True statement? No.

3. **For \( x = 11 - \sqrt{7} \)**:
- Substituting:
\[
(11 - \sqrt{7} - 7)^2 = (4 - \sqrt{7})^2 = 16 - 8\sqrt{7} + 7 = 23 - 8\sqrt{7} \neq 11
\]
- True statement? No.

4. **For \( x = 7 + \sqrt{11} \)**:
- Substituting:
\[
(7 + \sqrt{11} - 7)^2 = (\sqrt{11})^2 = 11
\]
- True statement? Yes.

5. **For \( x = 11 + \sqrt{7} \)**:
- Substituting:
\[
(11 + \sqrt{7} - 7)^2 = (4 + \sqrt{7})^2 = 16 + 8\sqrt{7} + 7 = 23 + 8\sqrt{7} \neq 11
\]
- True statement? No.

### Completing the Table

Now we can fill in the table:

\[
\begin{tabular}{|c|c|c|c|}
\hline
\( x \) & Substituted & Evaluate & True Statement? \\
\hline
\( 7-\sqrt{11} \) & \( (7-\sqrt{11}-7)^2 \) & \( 11 \) & Yes \\
\hline
7 & \( (7-7)^2 \) & \( 0 \) & No \\
\hline
\( 11-\sqrt{7} \) & \( (11-\sqrt{7}-7)^2 \) & \( 23 - 8\sqrt{7} \) & No \\
\hline
\( 7+\sqrt{11} \) & \( (7+\sqrt{11}-7)^2 \) & \( 11 \) & Yes \\
\hline
\( 11+\sqrt{7} \) & \( (11+\sqrt{7}-7)^2 \) & \( 23 + 8\sqrt{7} \) & No \\
\hline
\end{tabular}
\]

### Solutions

The smaller of the two solutions is:
\[
x = 7 - \sqrt{11}
\]
The larger of the two solutions is:
\[
x = 7 + \sqrt{11}
\]

Thus:
- The smaller solution is \( x = 7 - \sqrt{11} \).
- The larger solution is \( x = 7 + \sqrt{11} \).