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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page 15 of 16

Answer 1

To find the solutions for the quadratic equation \((x - 7)^2 = 11\), we’ll first solve the equation for \(x\).

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

2. **Now, solve for \(x\)**:
- Adding 7 to both sides gives two separate solutions:
\[
x = 7 + \sqrt{11}
\]
\[
x = 7 - \sqrt{11}
\]

Next, we need to evaluate these values and fill out your table.

- **Calculate \(7 + \sqrt{11}\)**:
\(\sqrt{11} \approx 3.3166\) (rounded)
Therefore,
\[
7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166
\]

- **Calculate \(7 - \sqrt{11}\)**:
\[
7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834
\]

Now let's identify the two solutions:
- **Smaller solution**: \(x = 7 - \sqrt{11} \approx 3.6834\)
- **Larger solution**: \(x = 7 + \sqrt{11} \approx 10.3166\)

Finally, placing the results in the answer format:
- The smaller of the two solutions is \(x = 7 - \sqrt{11}\).
- The larger of the two solutions is \(x = 7 + \sqrt{11}\).

So,
- The smaller of the two solutions is \(x \approx 3.68\).
- The larger of the two solutions is \(x \approx 10.32\).