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

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

{10,−22}
left brace 10 comma negative 22 right brace

{−2,−10}

Answer 1

To solve the equation \((x + 6)^2 = 8\) using square roots, we start by taking the square root of both sides:

\[
x + 6 = \pm\sqrt{8}
\]

The square root of 8 can be simplified:

\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]

So we have:

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

Now, we will solve for \(x\) in each case.

1. For \(x + 6 = 2\sqrt{2}\):
\[
x = 2\sqrt{2} - 6
\]

2. For \(x + 6 = -2\sqrt{2}\):
\[
x = -2\sqrt{2} - 6
\]

Now let's calculate the approximate values:

1. Calculating \(2\sqrt{2}\):
\[
2\sqrt{2} \approx 2 \times 1.414 \approx 2.828
\]
Therefore,
\[
x \approx 2.828 - 6 \approx -3.172
\]

2. Calculating \(-2\sqrt{2}\):
\[
-2\sqrt{2} \approx -2.828
\]
Therefore,
\[
x \approx -2.828 - 6 \approx -8.828
\]

So, the approximate solutions are:

\[
x \approx -3.17 \quad \text{and} \quad x \approx -8.83
\]

None of the given responses match these exact values, so please check if the responses provided match exact calculations or if there’s any additional context to consider. The approximate answers or any errors in the choices may have resulted from rounding or other discrepancies.

If we were to reconsider the options, the closest to our approximate solutions indicate the two solutions `{-3.17, -8.83}`, which do not exactly match with any provided.

If you have further details or context about the question, I'd be happy to assist further!