Question

Use the table to answer the question.

x Substituted Evaluate True Statement?
7−11−−√

7
11−7–√

7+11−−√

11+7–√

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=
.
Remaining Attempts : 3
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Answer 1

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 \(x\):
1. For the positive case:
\[
x - 7 = \sqrt{11} \implies x = 7 + \sqrt{11}
\]

2. 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:

1. **\(7 - \sqrt{11}\)**:
- Substituted: \(7 - \sqrt{11}\)
- Evaluate: \((7 - \sqrt{11} - 7)^2 = (-\sqrt{11})^2 = 11\)
- True statement? Yes, it satisfies the equation.

2. **\(7\)**:
- Substituted: \(7\)
- Evaluate: \((7 - 7)^2 = 0\), which is not equal to \(11\)
- True statement? No, it does not satisfy the equation.

3. **\(11 - 7\)** (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.

4. **\(7 + \sqrt{11}\)**:
- Substituted: \(7 + \sqrt{11}\)
- Evaluate: \((7 + \sqrt{11} - 7)^2 = (\sqrt{11})^2 = 11\)
- True statement? Yes, it satisfies the equation.

5. **\(11 + 7\)** (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 \(x = 7 + \sqrt{11}\).