Question

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

{−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

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

{−2,−10}

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 \(\sqrt{8}\):
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]

3. Therefore, we have two equations to solve:
\[
x + 6 = 2\sqrt{2} \quad \text{and} \quad x + 6 = -2\sqrt{2}
\]

4. Solve each equation for \(x\):

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

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

5. Evaluate \(2\sqrt{2}\):
\[
\sqrt{2} \approx 1.414 \quad \Rightarrow \quad 2\sqrt{2} \approx 2.828
\]

6. Compute the solutions:
\[
x \approx 2.828 - 6 = -3.172
\]
\[
x \approx -2.828 - 6 = -8.828
\]

The approximate solutions are:

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

Now check the options given:

None of the given options, {−1.17,−6.83}, {10,−22}, {−0.34,−11.66}, {−2,−10}, seem to directly match the calculations, but the approximate values can suggest a hypothetical misalignment.

To best match which pair of values represent this quadratic equation:

1. Round \( -3.172 \) to verify against the given sets. Although, \( -8.828 \) is not closely represented at all.
2. Finding that the correct pairs overall remain quickly defined rather than definitively settled.

In conclusion, the answer is not explicitly in the provided options, yet you may mean to resolve the quadratic tabulations through these extensions.

Final commentary: The answers can be approximated further if guided correctly, with refinement needed to indicate more accurately than current states reflect.