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

Question

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 \(-8.828\).