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

Question

Responses

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

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

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

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

Answer 1

To solve the quadratic equation \((x + 6)^2 = 8\), we can start by taking the square root of both sides.

1. Start with the equation: \[ (x + 6)^2 = 8 \]

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

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

4. Therefore, we have two cases:

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

5. Solve for \(x\) in both cases:

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

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

6. Now, compute the approximate values of these solutions:

   • Using \(\sqrt{2} \approx 1.414\):

     • \(2\sqrt{2} \approx 2 \times 1.414 \approx 2.828\),
     • Thus, \(x \approx 2.828 - 6 \approx -3.172\).

   • For the second case:

     • \(-2\sqrt{2} \approx -2.828\),
     • Thus, \(x \approx -2.828 - 6 \approx -8.828\).

The approximate solutions are \(x \approx -3.172\) and \(x \approx -8.828\).

None of the provided response options seem to match the approximate values we found directly. Let's take a look again at the given answers and see if any can correspond based on approximate evaluation.

None of the response options match this result, so it appears there may be a misunderstanding or error in the provided choices.

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