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=
x
=
. The larger of the two solutions is x=
x
=
.

Answer 1

To solve the quadratic equation \((x-7)^2 = 11\), we first need to find the solutions by taking the square root of both sides.

Step 1: Rewrite the equation \[ (x - 7)^2 = 11 \]

Step 2: Take the square root of both sides \[ x - 7 = \pm \sqrt{11} \]

Step 3: Solve for \(x\)

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

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

Now, we have our two solutions:

1. \(x = 7 + \sqrt{11}\) (approximately 10.32)
2. \(x = 7 - \sqrt{11}\) (approximately 3.68)

Identifying the smaller and larger solutions:

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

Thus:

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

Answers:

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

Answer 2

whats the two number answers

Answer 3

To provide the two numerical answers for \(x = 7 - \sqrt{11}\) and \(x = 7 + \sqrt{11}\), we can calculate them as follows:

1. Calculate \(7 - \sqrt{11}\):

   • The approximate value of \(\sqrt{11} \approx 3.3166\).
   • Therefore, \(7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834\).

2. Calculate \(7 + \sqrt{11}\):

   • Continuing with \(\sqrt{11} \approx 3.3166\).
   • Thus, \(7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166\).

So, rounded to two decimal points:

• The smaller solution is approximately \(3.68\).
• The larger solution is approximately \(10.32\).

Final answers:

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