Question

Solution Sets of Quadratic Equations Practice
Complete this assessment to review what you’ve learned. It will not count toward your grade.
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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'll first find the solutions by isolating \(x\).

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

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

- For the negative case:
\[
x - 7 = -\sqrt{11} \implies x = 7 - \sqrt{11}
\]

So the two solutions to the equation are:
- \(x_1 = 7 - \sqrt{11}\) (the smaller solution)
- \(x_2 = 7 + \sqrt{11}\) (the larger solution)

Now, let's fill in the table with the necessary evaluations.

- **Substituting \(7 - \sqrt{11}\)**:
\[
(7 - \sqrt{11}) - 7 = -\sqrt{11}
\]
Squaring this gives:
\[
(-\sqrt{11})^2 = 11 \quad \text{(True statement)}
\]

- **Substituting \(7\)**:
\[
7 - 7 = 0
\]
Squaring this gives:
\[
0^2 = 0 \quad \text{(not equal to 11, False statement)}
\]

- **Substituting \(7 + \sqrt{11}\)**:
\[
(7 + \sqrt{11}) - 7 = \sqrt{11}
\]
Squaring this gives:
\[
(\sqrt{11})^2 = 11 \quad \text{(True statement)}
\]

- **Substituting \(11 - 7\)** (which is \(4\)):
\[
4 - 7 = -3
\]
Squaring this gives:
\[
(-3)^2 = 9 \quad \text{(not equal to 11, False statement)}
\]

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

Finally, the answers are:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).