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

Question

.(1 point)
Responses

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

{−1.17,−6.83}

Answer 1

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

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

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

3. Therefore, we can write: \[ x + 6 = 2\sqrt{2} \quad \text{or} \quad x + 6 = -2\sqrt{2} \]

4. Solve for \(x\):

   • For the first case: \[ x = 2\sqrt{2} - 6 \]
   • For the second case: \[ x = -2\sqrt{2} - 6 \]

5. Calculate the numerical values for better understanding:

   • Approximate \(2\sqrt{2} \approx 2 \cdot 1.414 \approx 2.828\): \[ x \approx 2.828 - 6 \approx -3.172 \]
   • Approximate \(-2\sqrt{2} \approx -2.828\): \[ x \approx -2.828 - 6 \approx -8.828 \]

Thus, the approximate solutions are: \[ x \approx -3.17 \quad \text{and} \quad x \approx -8.83 \]

However, given the original options, it seems there's a rounding difference. The direct numerical representation does not match any of the given ranges exactly.

Please clarify or double-check the options provided. If you'd like to find exact values, our earlier results give us: \[ x = -6 \pm 2\sqrt{2} \]

In conclusion, based on the form of the answers you've provided, none appear to match the calculations from the specified quadratic equation correctly.