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Original Question
Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x−6)^2=51Question
Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: ((x+16)^2/9) =7
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Answers
Answer
i can only use one solution
Answer
incorrect
Answered by
GPT-4o mini
To solve the quadratic equation \(\frac{(x+16)^2}{9} = 7\), we follow these steps:
1. **Eliminate the fraction**: Multiply both sides by 9 to get rid of the denominator:
\[
(x + 16)^2 = 7 \cdot 9
\]
\[
(x + 16)^2 = 63
\]
2. **Take the square root of both sides**:
\[
x + 16 = \pm \sqrt{63}
\]
3. **Calculate \(\sqrt{63}\)**:
\(\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}\). We can approximate \(\sqrt{7} \approx 2.6457513110645906\), so:
\[
3\sqrt{7} \approx 3 \cdot 2.6457513110645906 \approx 7.937253933193776
\]
Rounding this to the nearest hundredth gives approximately \(7.94\).
4. **Solve for \(x\)**: We have two equations to solve:
\[
x + 16 = 7.94 \quad \text{and} \quad x + 16 = -7.94
\]
- For the first equation:
\[
x + 16 = 7.94 \implies x = 7.94 - 16 \implies x \approx -8.06
\]
- For the second equation:
\[
x + 16 = -7.94 \implies x = -7.94 - 16 \implies x \approx -23.94
\]
5. **Final answers**:
The solutions to the equation \(\frac{(x+16)^2}{9} = 7\) are:
\[
x \approx -8.06 \quad \text{and} \quad x \approx -23.94
\]
1. **Eliminate the fraction**: Multiply both sides by 9 to get rid of the denominator:
\[
(x + 16)^2 = 7 \cdot 9
\]
\[
(x + 16)^2 = 63
\]
2. **Take the square root of both sides**:
\[
x + 16 = \pm \sqrt{63}
\]
3. **Calculate \(\sqrt{63}\)**:
\(\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}\). We can approximate \(\sqrt{7} \approx 2.6457513110645906\), so:
\[
3\sqrt{7} \approx 3 \cdot 2.6457513110645906 \approx 7.937253933193776
\]
Rounding this to the nearest hundredth gives approximately \(7.94\).
4. **Solve for \(x\)**: We have two equations to solve:
\[
x + 16 = 7.94 \quad \text{and} \quad x + 16 = -7.94
\]
- For the first equation:
\[
x + 16 = 7.94 \implies x = 7.94 - 16 \implies x \approx -8.06
\]
- For the second equation:
\[
x + 16 = -7.94 \implies x = -7.94 - 16 \implies x \approx -23.94
\]
5. **Final answers**:
The solutions to the equation \(\frac{(x+16)^2}{9} = 7\) are:
\[
x \approx -8.06 \quad \text{and} \quad x \approx -23.94
\]
Answered by
GPT-4o mini
If you can only use one solution, you can choose either of the two solutions we found.
The solutions to the equation \( \frac{(x+16)^2}{9} = 7 \) were:
1. \( x \approx -8.06 \)
2. \( x \approx -23.94 \)
If you need to select one solution, you can choose the one that best fits your requirements (e.g., closer to a certain value, or a specific context for your problem).
For example, if you are simply looking for the solution with the smaller absolute value, you would choose:
\[
\boxed{-8.06}
\]
If you have any further specifications, please let me know!
The solutions to the equation \( \frac{(x+16)^2}{9} = 7 \) were:
1. \( x \approx -8.06 \)
2. \( x \approx -23.94 \)
If you need to select one solution, you can choose the one that best fits your requirements (e.g., closer to a certain value, or a specific context for your problem).
For example, if you are simply looking for the solution with the smaller absolute value, you would choose:
\[
\boxed{-8.06}
\]
If you have any further specifications, please let me know!
Answered by
GPT-4o mini
I apologize for any confusion earlier. Let's re-solve the equation step by step and check our results properly.
Given the equation:
\[
\frac{(x + 16)^2}{9} = 7
\]
1. **Multiply both sides by 9 to eliminate the fraction**:
\[
(x + 16)^2 = 63
\]
2. **Take the square root of both sides**:
\[
x + 16 = \pm \sqrt{63}
\]
The square root of \(63\) can be approximated. First, we can simplify:
\[
\sqrt{63} \approx 7.937 (rounded to the nearest hundredth is 7.94)
\]
So, we have two equations to solve:
\[
x + 16 = 7.94 \quad \text{and} \quad x + 16 = -7.94
\]
3. **Solve for \(x\)**:
- From \(x + 16 = 7.94\):
\[
x = 7.94 - 16 \implies x \approx -8.06
\]
- From \(x + 16 = -7.94\):
\[
x = -7.94 - 16 \implies x \approx -23.94
\]
If you are looking for the correct answer regarding which solution is acceptable or preferred, let’s evaluate both.
- The first solution \(x \approx -8.06\) could be seen as more typical for real-world applications, being closer to zero.
If I misunderstood your question about choosing "the" one solution, please clarify the criteria for selecting the final solution, and I would be glad to assist further. Otherwise, if you just want one of the results, most contexts would typically use:
\[
\boxed{-8.06}
\]
if you're looking for the solution that is numerically closer to zero.
Given the equation:
\[
\frac{(x + 16)^2}{9} = 7
\]
1. **Multiply both sides by 9 to eliminate the fraction**:
\[
(x + 16)^2 = 63
\]
2. **Take the square root of both sides**:
\[
x + 16 = \pm \sqrt{63}
\]
The square root of \(63\) can be approximated. First, we can simplify:
\[
\sqrt{63} \approx 7.937 (rounded to the nearest hundredth is 7.94)
\]
So, we have two equations to solve:
\[
x + 16 = 7.94 \quad \text{and} \quad x + 16 = -7.94
\]
3. **Solve for \(x\)**:
- From \(x + 16 = 7.94\):
\[
x = 7.94 - 16 \implies x \approx -8.06
\]
- From \(x + 16 = -7.94\):
\[
x = -7.94 - 16 \implies x \approx -23.94
\]
If you are looking for the correct answer regarding which solution is acceptable or preferred, let’s evaluate both.
- The first solution \(x \approx -8.06\) could be seen as more typical for real-world applications, being closer to zero.
If I misunderstood your question about choosing "the" one solution, please clarify the criteria for selecting the final solution, and I would be glad to assist further. Otherwise, if you just want one of the results, most contexts would typically use:
\[
\boxed{-8.06}
\]
if you're looking for the solution that is numerically closer to zero.
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