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

Question

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

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

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

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

​ {−0.34,−11.66}

Answer 1

To solve the equation \((x + 6)^2 = 8\) using square roots, follow the steps below:

1. Take the square root of both sides: \[ x + 6 = \pm \sqrt{8} \] Simplifying \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \] Thus, we have: \[ x + 6 = \pm 2\sqrt{2} \]

2. Isolate \(x\): \[ x = -6 \pm 2\sqrt{2} \]

3. Calculate the two possible values for \(x\):

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

   Approximating \(2\sqrt{2} \approx 2 \times 1.414 \approx 2.828\):

   • \(x \approx -6 + 2.828 \approx -3.172\)
   • \(x \approx -6 - 2.828 \approx -8.828\)

4. Expressing in set notation: The approximate solutions are: \[ x_1 \approx -3.172 \] \[ x_2 \approx -8.828 \] Thus the solutions can be approximately represented as: {−3.17, −8.83}

However, if we don't approximate the values, we can write the exact solutions as: \[ \left{-6 + 2\sqrt{2}, -6 - 2\sqrt{2}\right} \]

None of the provided options match exactly.

However, if we calculate the numerical values of \( -6 + 2\sqrt{2} \) and \( -6 - 2\sqrt{2} \), they will likely fall within a range that matches one of the provided sets. Thus it seems like the final answer with numerical approximations is:

\[ {−3.17, −8.83} \]

Since the exact options provided do not match, none respond correctly to this computation.