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

{−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

{10,−22}

Answer 1

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

1. Take the square root of both sides:
\[
x + 6 = \pm \sqrt{8}
\]
Since \(\sqrt{8} = 2\sqrt{2}\), we can rewrite the equation as:
\[
x + 6 = \pm 2\sqrt{2}
\]

2. Now, we can solve for \(x\) by isolating it:
\[
x = -6 \pm 2\sqrt{2}
\]

3. This gives us two solutions:
\[
x_1 = -6 + 2\sqrt{2}
\]
\[
x_2 = -6 - 2\sqrt{2}
\]

4. Now we will evaluate \(2\sqrt{2}\) using the approximate value of \(\sqrt{2} \approx 1.414\):
\[
2\sqrt{2} \approx 2 \times 1.414 = 2.828
\]

5. Calculating the two values:
\[
x_1 = -6 + 2.828 \approx -3.172
\]
\[
x_2 = -6 - 2.828 \approx -8.828
\]

None of these results match the given options exactly. However, we need to check whether we can find a more precise numeric output.

The two solutions correspond approximately to numerical values we calculated, and we should also consider the exact square root values:

- When computed accurately:
- \( x_1 = -6 + 2\sqrt{2} \) (approximately -3.172)
- \( x_2 = -6 - 2\sqrt{2} \) (approximately -8.828)

Looking closer, if we evaluate \(-2\) and \(-10\):
- Evaluating approximate values \(2\sqrt{4}\) through recalculating using algebra.

The correct decimal places given do not appear here.

Returning to how you might best choose an answer without calculation errors occurring with square roots:
Among the closest estimates, evaluating pairs aligned numerically yields:
- \( \{-2, -10\}\) determining for estimates rounding down to usable base equations.

Thus, the closest valid solution option is:
**{−2,−10}.**