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= .

Answer 1

To solve the quadratic equation \((x - 7)^2 = 11\), we will first find the solutions by following these steps:

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

2. **Add 7 to both sides**:
\[
x = 7 \pm \sqrt{11}
\]

Now, we have two potential solutions:
- The smaller solution is:
\[
x = 7 - \sqrt{11}
\]
- The larger solution is:
\[
x = 7 + \sqrt{11}
\]

Next, we need to identify these two solutions numerically. Since \(\sqrt{11} \approx 3.32\):
- Smaller solution:
\[
x = 7 - \sqrt{11} \approx 7 - 3.32 \approx 3.68
\]

- Larger solution:
\[
x = 7 + \sqrt{11} \approx 7 + 3.32 \approx 10.32
\]

Thus, we can conclude:
- The smaller of the two solutions \(x = 7 - \sqrt{11}\).
- The larger of the two solutions \(x = 7 + \sqrt{11}\).

Finally, we can fill in the blanks:
- The smaller of the two solutions is \( x = 7 - \sqrt{11} \).
- The larger of the two solutions is \( x = 7 + \sqrt{11} \).

In summary:
- The smaller solution is \( x \approx 3.68 \).
- The larger solution is \( x \approx 10.32 \).

Note: If you want to express the exact solutions without decimal approximation:
- The smaller solution is \( x = 7 - \sqrt{11} \).
- The larger solution is \( x = 7 + \sqrt{11} \).