Question
Solve the following quadratic equation using square roots: (x+6)22=8 .(1 point)
Responses
{−0.34,−11.66}
left brace negative 0.34 comma negative 11.66 right brace
{10,−22}
left brace 10 comma negative 22 right brace
{−2,−10}
left brace negative 2 comma negative 10 right brace
{−1.17,−6.83}
Responses
{−0.34,−11.66}
left brace negative 0.34 comma negative 11.66 right brace
{10,−22}
left brace 10 comma negative 22 right brace
{−2,−10}
left brace negative 2 comma negative 10 right brace
{−1.17,−6.83}
Answers
GPT-4o mini
To solve the quadratic equation \((x + 6)^2 = 8\) using square roots, follow these steps:
1. **Take the square root of both sides**:
\[
x + 6 = \pm \sqrt{8}
\]
2. **Simplify \(\sqrt{8}\)**:
\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
\]
So the equation becomes:
\[
x + 6 = \pm 2\sqrt{2}
\]
3. **Solve for \(x\)**:
\[
x = -6 \pm 2\sqrt{2}
\]
4. **Calculate the two possible solutions**:
- For the positive case:
\[
x = -6 + 2\sqrt{2}
\]
- For the negative case:
\[
x = -6 - 2\sqrt{2}
\]
5. **Approximate the values**:
Using \(\sqrt{2} \approx 1.414\):
- For \(x = -6 + 2\sqrt{2}\):
\[
x \approx -6 + 2(1.414) \approx -6 + 2.828 \approx -3.172
\]
- For \(x = -6 - 2\sqrt{2}\):
\[
x \approx -6 - 2(1.414) \approx -6 - 2.828 \approx -8.828
\]
Now, the solutions can be approximated to around \(-3.17\) and \(-8.83\).
**Final Answers**: The approximate solutions are \(-3.17\) and \(-8.83\).
Since none of your provided choices match these exact outputs, please verify if 2 is rounded from somewhere or if I need to return to check other questions. However, further checking may suggest a re-evaluation on approximating square roots.
1. **Take the square root of both sides**:
\[
x + 6 = \pm \sqrt{8}
\]
2. **Simplify \(\sqrt{8}\)**:
\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
\]
So the equation becomes:
\[
x + 6 = \pm 2\sqrt{2}
\]
3. **Solve for \(x\)**:
\[
x = -6 \pm 2\sqrt{2}
\]
4. **Calculate the two possible solutions**:
- For the positive case:
\[
x = -6 + 2\sqrt{2}
\]
- For the negative case:
\[
x = -6 - 2\sqrt{2}
\]
5. **Approximate the values**:
Using \(\sqrt{2} \approx 1.414\):
- For \(x = -6 + 2\sqrt{2}\):
\[
x \approx -6 + 2(1.414) \approx -6 + 2.828 \approx -3.172
\]
- For \(x = -6 - 2\sqrt{2}\):
\[
x \approx -6 - 2(1.414) \approx -6 - 2.828 \approx -8.828
\]
Now, the solutions can be approximated to around \(-3.17\) and \(-8.83\).
**Final Answers**: The approximate solutions are \(-3.17\) and \(-8.83\).
Since none of your provided choices match these exact outputs, please verify if 2 is rounded from somewhere or if I need to return to check other questions. However, further checking may suggest a re-evaluation on approximating square roots.