Question

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 equation \((x - 7)^2 = 11\), we first need to find the values of \(x\) that satisfy this equation.

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

2. **Solving for \(x\)**:
For the positive root:
\[
x - 7 = \sqrt{11} \implies x = 7 + \sqrt{11}
\]
For the negative root:
\[
x - 7 = -\sqrt{11} \implies x = 7 - \sqrt{11}
\]

3. **Finding the approximate values of the solutions**:
The value of \(\sqrt{11}\) is approximately 3.32.

So the two solutions are:
\[
x = 7 + \sqrt{11} \approx 7 + 3.32 \approx 10.32
\]
\[
x = 7 - \sqrt{11} \approx 7 - 3.32 \approx 3.68
\]

Now we can fill out the table with the evaluations of \(x = 7\), \(x = 11\), \(x = 7 + \sqrt{11}\), and \(x = 7 - \sqrt{11}\):

### Table Completion:
1. Substitute \(x = 7\):
\[
(7 - 7)^2 = 0 \quad (\text{not a solution})
\]

2. Substitute \(x = 11\):
\[
(11 - 7)^2 = 4 \quad (\text{not a solution})
\]

3. Substitute \(x = 7 + \sqrt{11}\):
\[
((7 + \sqrt{11}) - 7)^2 = (\sqrt{11})^2 = 11 \quad (\text{solution})
\]

4. Substitute \(x = 7 - \sqrt{11}\):
\[
((7 - \sqrt{11}) - 7)^2 = (-\sqrt{11})^2 = 11 \quad (\text{solution})
\]

### Conclusion:
The two solutions to the equation \((x - 7)^2 = 11\) are:
- The smaller solution is \(x = 7 - \sqrt{11}\)
- The larger solution is \(x = 7 + \sqrt{11}\)

### Final Answer:
The smaller of the two solutions is \(x = 7 - \sqrt{11}\). The larger of the two solutions is \(x = 7 + \sqrt{11}\).