Use the table to answer the question.x Substituted Evaluate True Statement?7−11−−√711−7–√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 \((x - 7)^2 = 11\), we first take the square root of both sides:

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

This gives us two equations to solve:

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

Now, we can solve each equation for \(x\):

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

\[ x = 7 + \sqrt{11} \]

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

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

Now we have our two solutions:

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

Now let's identify the values explicitly

Given that \(\sqrt{11} \approx 3.32\) (approximately):

1. Smaller solution:
   \[ x = 7 - \sqrt{11} \approx 7 - 3.32 = 3.68 \]

2. Larger solution:
   \[ x = 7 + \sqrt{11} \approx 7 + 3.32 = 10.32 \]

Thus, the final answers for the two solutions to the quadratic equation are:

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