Question

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:

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

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

Now we need to identify and complete the table with the provided values and check if they satisfy the equation.

Let’s evaluate:

1. **Value \(7 - \sqrt{11}\):**
\[
(7 - \sqrt{11} - 7)^2 = (-\sqrt{11})^2 = 11 \quad \text{(valid solution)}
\]

2. **Value \(7\):**
\[
(7 - 7)^2 = 0 \neq 11 \quad \text{(not a solution)}
\]

3. **Value \(11 - \sqrt{7}\):**
\[
(11 - \sqrt{7} - 7)^2 = (4 - \sqrt{7})^2 \neq 11 \quad \text{(not a solution)}
\]

4. **Value \(7 + \sqrt{11}\):**
\[
(7 + \sqrt{11} - 7)^2 = (\sqrt{11})^2 = 11 \quad \text{(valid solution)}
\]

5. **Value \(11 + \sqrt{7}\):**
\[
(11 + \sqrt{7} - 7)^2 = (4 + \sqrt{7})^2 \neq 11 \quad \text{(not a solution)}
\]

Now, we have identified the valid solutions to the equation \( (x - 7)^2 = 11\):

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