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=.
Answers
GPT-4o mini
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 \(x \approx 10.3166\).
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 \(x \approx 10.3166\).
r
wrong
GPT-4o mini
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.
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.