Question

Use the table to answer the 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 first need to find the solutions by taking the square root of both sides. Here are the steps:

1. **Rewrite the equation**:
\[
(x-7)^2 = 11
\]

2. **Take the square root of both sides**:
\[
x - 7 = \pm \sqrt{11}
\]

3. **Solve for \(x\)**:
\[
x = 7 + \sqrt{11} \quad \text{and} \quad x = 7 - \sqrt{11}
\]

Now we'll compute the numeric values of each solution.

- For \(x = 7 + \sqrt{11}\):
- \(\sqrt{11} \approx 3.317\) (approximately)
- Thus, \(x \approx 7 + 3.317 = 10.317\)

- For \(x = 7 - \sqrt{11}\):
- Thus, \(x \approx 7 - 3.317 = 3.683\)

Now we can state the two solutions:

- The smaller solution is:
\[
x \approx 3.683
\]

- The larger solution is:
\[
x \approx 10.317
\]

### In conclusion:
- The smaller of the two solutions is \(x = 7 - \sqrt{11} \approx 3.683\).
- The larger of the two solutions is \(x = 7 + \sqrt{11} \approx 10.317\).

**Final answers:**
- The smaller of the two solutions is \(x = 7 - \sqrt{11} \approx 3.683\).
- The larger of the two solutions is \(x = 7 + \sqrt{11} \approx 10.317\).