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

Solve the following quadratic equation using square roots: (x+6)^2/2=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 2

To solve the equation \( (x + 6)^2 = 8 \) using square roots, we can 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, we have:
\[
x + 6 = \pm 2\sqrt{2}
\]

3. Now, we can solve for \(x\) by isolating it:
\[
x = -6 \pm 2\sqrt{2}
\]

4. This gives us two potential solutions:
\[
x = -6 + 2\sqrt{2} \quad \text{and} \quad x = -6 - 2\sqrt{2}
\]

5. Now, we can calculate the numerical values.

- For \( x = -6 + 2\sqrt{2} \):
\[
\approx -6 + 2 \times 1.414 \approx -6 + 2.828 = -3.172
\]

- For \( x = -6 - 2\sqrt{2} \):
\[
\approx -6 - 2 \times 1.414 \approx -6 - 2.828 = -8.828
\]

6. Therefore, the solutions in decimal form are approximately:
\[
x \approx -3.17 \quad \text{and} \quad x \approx -8.83
\]

Comparing the approximate values to the given responses, none of the options match the values calculated above. So, let's confirm the calculations.

Lastly, simplifying:
- The negative solution \( -6 - 2\sqrt{2} \) is approximately \( -8.83 \).
- The positive solution \( -6 + 2\sqrt{2} \) is approximately \( -3.17 \).

The correct approximate values \( \{-3.17, -8.83\} \) do not appear in the response options provided, implying that the question options may have different representations or calculations that require double-checking.

Please validate the responses against the original question's constraints or clarify if the responses listed are meant to include different reference points or approximations.

Answer 3

To solve the equation \(\frac{(x + 6)^2}{2} = 8\), we can follow these steps:

1. **Multiply both sides by 2** to eliminate the fraction:
\[
(x + 6)^2 = 16
\]

2. **Take the square root of both sides**:
\[
x + 6 = \pm 4
\]

3. **Solve for \(x\)**:
- For the positive root:
\[
x + 6 = 4 \implies x = 4 - 6 = -2
\]
- For the negative root:
\[
x + 6 = -4 \implies x = -4 - 6 = -10
\]

Thus, the two solutions are:
\[
x = -2 \quad \text{and} \quad x = -10
\]

So, the final solutions are \(\{-2, -10\}\).

From the options given:
- \(\{-2, -10\}\)

This matches with one of the provided choices! Thus the correct answer is:
\[
\{-2, -10\}
\]