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 equation \((x - 7)^2 = 11\), we first need to isolate \(x\).

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

2. Solve for \(x\): \[ x = 7 + \sqrt{11} \quad \text{or} \quad x = 7 - \sqrt{11} \]

Now, we need to evaluate and determine the approximate values of these solutions.

• The approximate value of \(\sqrt{11}\) is about \(3.3166\).

Thus, the solutions become:

1. \(x = 7 + \sqrt{11} \approx 7 + 3.3166 = 10.3166\)
2. \(x = 7 - \sqrt{11} \approx 7 - 3.3166 = 3.6834\)

Now, identify the two solutions:

• The smaller of the two solutions is: \[ x \approx 7 - \sqrt{11} \implies x \approx 3.6834 \]
• The larger of the two solutions is: \[ x \approx 7 + \sqrt{11} \implies x \approx 10.3166 \]

Answers:

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

Answer 2

wrong

Answer 3

Let’s carefully solve the equation \((x - 7)^2 = 11\) step-by-step to find the solutions correctly.

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

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

3. Solve for \(x\):

   • For the positive square root: \[ x = 7 + \sqrt{11} \]
   • For the negative square root: \[ x = 7 - \sqrt{11} \]

4. Now let's calculate the approximate numerical values:

   • The approximate value of \(\sqrt{11}\) is about \(3.3166\).
   • Calculating the two solutions:
     • Larger solution: \[ x = 7 + \sqrt{11} \approx 7 + 3.3166 \approx 10.3166 \]
     • Smaller solution: \[ x = 7 - \sqrt{11} \approx 7 - 3.3166 \approx 3.6834 \]

Final Answers:

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

These calculations are correct. If you have other specific values you want me to evaluate from a table you haven't provided, please share those, and I can evaluate them for you.