Question

Solve the following quadratic equation using square roots: (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 ​ {−0.34,−11.66} ​ left brace negative 0.34 comma negative 11.66 right brace {10,−22}

Answer 1

To solve the quadratic equation \((x + 6)^2 = 8\), we will first take the square root of both sides.

1. Start with the equation:
\[
(x + 6)^2 = 8
\]

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

3. Simplify \(\sqrt{8}\):
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]
So, we have:
\[
x + 6 = \pm 2\sqrt{2}
\]

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

Now we will compute the two possible values of \(x\):

1. First value:
\[
x = -6 + 2\sqrt{2}
\]

2. Second value:
\[
x = -6 - 2\sqrt{2}
\]

Using a calculator for approximate values:
- \(\sqrt{2} \approx 1.414\)
- Then, approximate values:
\[
x \approx -6 + 2(1.414) \approx -6 + 2.828 \approx -3.172
\]

\[
x \approx -6 - 2(1.414) \approx -6 - 2.828 \approx -8.828
\]

From our calculations using a calculator, we obtain two values:
- \(x \approx -3.17\)
- \(x \approx -8.83\)

After looking at the answer options:

- None of the given options {−2,−10}, {−1.17,−6.83}, {−0.34,−11.66}, {10,−22} match our values.

However, we can verify our steps:
1. If we instead approximate \(\sqrt{8} = 2.828\).
- This confirms our two potential approximate values are not present in your responses.

Thus, if the given answer choices do not match the expected results of this calculation, please check the initial problem statement for accuracy or ensure that the alternatives are complete based on the quadratic values calculated.

The correct solutions in terms of proximity aren't listed among the choices that were provided. The solutions \(x \approx -3.17\) and \(x \approx -8.83\) rounded do not yield appropriate selections as per available options.