Use the table to answer the question.

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 quadratic equation \((x - 7)^2 = 11\), let's first take the square root of both sides:

\[ x - 7 = \sqrt{11} \quad \text{or} \quad x - 7 = -\sqrt{11} \]

Now we can solve for \(x\):

1. For the positive root: \[ x - 7 = \sqrt{11} \implies x = 7 + \sqrt{11} \]

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

So the two solutions to the quadratic equation are:

\[ x = 7 + \sqrt{11} \quad \text{and} \quad x = 7 - \sqrt{11} \]

Next, let's identify the numerical values of these solutions:

• \(7 - \sqrt{11}\) (this is the smaller solution).
• \(7 + \sqrt{11}\) (this is the larger solution).

To get approximate values, note that \(\sqrt{11} \approx 3.316\):

1. \(7 - \sqrt{11} \approx 7 - 3.316 \approx 3.684\)
2. \(7 + \sqrt{11} \approx 7 + 3.316 \approx 10.316\)

Using the phrasing in your question, we have:

The smaller of the two solutions is \(x = 7 - \sqrt{11} \approx 3.684\). The larger of the two solutions is \(x = 7 + \sqrt{11} \approx 10.316\).

Therefore:

• The smaller solution is \(x = 7 - \sqrt{11}\).
• The larger solution is \(x = 7 + \sqrt{11}\).