Solve the following quadratic equation using square roots: (x+6)22=8

Question

(
𝑥
+
6
)
2
2
=
8
.(1 point)
Responses

{−2,−10}
{
−
2
,
−
10
}
left brace negative 2 comma negative 10 right brace

​ {−0.34,−11.66}
{
−
0.34
,
−
11.66
}
​ left brace negative 0.34 comma negative 11.66 right brace

{−1.17,−6.83}
{
−
1.17
,
−
6.83
}
left brace negative 1.17 comma negative 6.83 right brace

{10,−22}

Answer 1

To solve the 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. Now, isolate \(x\) by subtracting 6 from both sides:

\[ x = -6 \pm 2\sqrt{2} \]

This gives us two solutions:

\[ x = -6 + 2\sqrt{2} \quad \text{and} \quad x = -6 - 2\sqrt{2} \]

4. Now, we can calculate the approximate numerical values of these solutions:

• For \(x = -6 + 2\sqrt{2}\):

\[ \approx -6 + 2(1.414) = -6 + 2.828 \approx -3.172 \]

• For \(x = -6 - 2\sqrt{2}\):

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

5. Thus, the approximate solutions are:

\[ { -3.17, -8.83 } \]

Based on the options you have provided, none exactly match our calculated solutions, as my approximations do not correspond to any of your listed answers.

It seems there was confusion in interpreting the given choices. However, if required, please verify the coefficients used in your original equation or if any details were omitted regarding the structure of the problem. If you meant \( (x+6)^2 = 8 \) after reviewing, this calculation will always hold, leading us through similarly structured numerical outputs.