Question

Solve the following quadratic equation using square roots: (x+6)22=8 .(1 point)
Responses

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

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

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

​ {−0.34,−11.66}

Answer 1

To solve the quadratic equation \((x + 6)^2 = 8\), we can take the following steps:

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

2. Simplify \(\sqrt{8}\):
\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
\]

3. Now, rewrite the equation:
\[
x + 6 = 2\sqrt{2} \quad \text{or} \quad x + 6 = -2\sqrt{2}
\]

4. Solve for \(x\) in both cases:

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

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

5. Now we will calculate the numerical values:

\(\sqrt{2} \approx 1.414\), so:
- \(2\sqrt{2} \approx 2 \times 1.414 \approx 2.828\)
- Therefore, \( x = 2.828 - 6 \approx -3.172 \)
- \( x = -2.828 - 6 \approx -8.828 \)

6. Thus, the approximate solutions are:
\[
\{ -3.172, -8.828 \}
\]

The given options do not match precisely with these calculations. We can look at the approximate decimal values and check the options provided:
1. {10, −22}
2. {−1.17, −6.83}
3. {−2, −10}
4. {−0.34, −11.66}

None of the provided responses are correct based on the calculations.

Please double-check your equation, or if "8" in the equation might have been another value. Otherwise, the correct process yields approximate solutions \(-3.172\) and \(-8.828\).