Question

Solve the following quadratic equation using square roots:%C2%A0 (x+6)22=8 .(1 point) Responses {−2,−10} left brace negative 2 comma negative 10 right brace {−1.17,−6.83} left brace negative 1.17 comma negative 6.83 right brace %E2%80%8B%C2%A0{−0.34,−11.66} %E2%80%8B%C2%A0 left brace negative 0.34 comma negative 11.66 right brace {10,−22}

Answer 1

To solve the quadratic equation \((x + 6)^2 = 8\) using square roots, we follow these steps:

1. **Take the square root of both sides**:
\[
x + 6 = \pm \sqrt{8}
\]

2. **Simplify \(\sqrt{8}\)**:
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]
Thus, we rewrite the equation as:
\[
x + 6 = \pm 2\sqrt{2}
\]

3. **Solve for \(x\)** by isolating it:
- For the positive root:
\[
x = -6 + 2\sqrt{2}
\]
- For the negative root:
\[
x = -6 - 2\sqrt{2}
\]

4. **Calculate the numerical values**:
- The approximate value of \(2\sqrt{2}\) is:
\[
2\sqrt{2} \approx 2 \times 1.414 \approx 2.828
\]
- Therefore, we can find:
- First root:
\[
x \approx -6 + 2.828 \approx -3.172
\]
- Second root:
\[
x \approx -6 - 2.828 \approx -8.828
\]

Since these approximations do not exactly match with the given options, let's reevaluate using exact values.

Using the values of \(-6 \pm 2\sqrt{2}\):
- If we convert \(2\sqrt{2}\) back into decimal values, we find:

1. \(x_1 = -6 + 2\sqrt{2} \approx -3.172\) (This approximates correctly).
2. \(x_2 = -6 - 2\sqrt{2} \approx -8.828\) (This approximates correctly).

Now, let's compare with the multiple choice options given in your question:

- Possible choices include:
- {−2,−10}
- {−1.17,−6.83}
- {−0.34,−11.66}
- {10,−22}

None of these provided multiple-choice answers match our calculations based on the \((x + 6)^2 = 8\) equation solution.

It looks like there may have been a mistake somewhere in either the calculations or in the provided options. Double-check the specific results necessary based on the options given or verify the choice.