Question

Use the table to answer the question.

x Substituted Evaluate True Statement?
7−11−−√
7
−
11

7
11−7–√
11
−
7

7+11−−√
7
+
11

11+7–√
11
+
7

For the quadratic equation (x−7)2=11
(
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=
𝑥
=
. The larger of the two solutions is x=
𝑥
=
.

Answer 1

To solve the quadratic equation \((x - 7)^2 = 11\), we first take the square root of both sides:

\[
x - 7 = \pm \sqrt{11}
\]

This leads to two equations:

1. \(x - 7 = \sqrt{11}\)
2. \(x - 7 = -\sqrt{11}\)

Now we solve for \(x\) in both cases:

1. \(x = 7 + \sqrt{11}\)
2. \(x = 7 - \sqrt{11}\)

Next, we can evaluate the approximate values of these two solutions:

- For \(x = 7 + \sqrt{11}\):
\(\sqrt{11} \approx 3.3166\)
So, \(x \approx 7 + 3.3166 \approx 10.3166\) (This is the larger solution.)

- For \(x = 7 - \sqrt{11}\):
\(x \approx 7 - 3.3166 \approx 3.6834\) (This is the smaller solution.)

Thus, the solutions to the quadratic equation are:

- The smaller of the two solutions is \(x \approx 3.6834\).
- The larger of the two solutions is \(x \approx 10.3166\).

Now, to provide the answers in the required format:

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