Solve the following quadratic equation using square roots: (x+6)22=8.(1 point)%0D%0AResponses%0D%0A%0D%0A{10,−22}%0D%0Aleft brace 10 comma negative 22 right brace%0D%0A%0D%0A{−2,−10}%0D%0Aleft brace negative 2 comma negative 10 right brace%0D%0A%0D%0A{−1.17,−6.83}%0D%0Aleft brace negative 1.17 comma negative 6.83 right brace%0D%0A%0D%0A%E2%80%8B {−0.34,−11.66}

Question

GPT-4 Turbo · Answer

First, let's correct the equation you've provided as there seem to be some formatting issues with the repeated "22". Let's assume you meant to write the equation as:
\[(x + 6)^2 = 8.\]

Now, we will solve for x using square roots:

1. Isolate the squared term by taking the square root of both sides of the equation:
\[ \sqrt{(x + 6)^2} = \sqrt{8}. \]

2. Simplify the square root of the squared term and the square root of 8:
\[ x + 6 = \pm\sqrt{8}. \]

Since \(\sqrt{8}\) simplifies to \(2\sqrt{2}\) (because \(8 = 4 \times 2\) and the square root of 4 is 2), we have:
\[ x + 6 = \pm2\sqrt{2}. \]

3. Subtract 6 from both sides to solve for x:
\[ x = -6 \pm 2\sqrt{2}. \]

4. Evaluate the two solutions by considering both the positive and negative square root:
\[ x_1 = -6 + 2\sqrt{2}, \]
\[ x_2 = -6 - 2\sqrt{2}. \]

You've provided a set of responses in decimal form, so let's convert \(2\sqrt{2}\) to a decimal. The approximate value of \(\sqrt{2}\) is 1.414, so:

\[ 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828. \]

Now we can approximate the two solutions:
\[ x_1 \approx -6 + 2.828 \approx -3.172, \]
\[ x_2 \approx -6 - 2.828 \approx -8.828. \]

Comparing these results with the response options you provided, none of them match exactly. However, rounding to two decimal places, the solutions would be approximately:

\[ x_1 \approx -3.17, \]
\[ x_2 \approx -8.83. \]

This does not match exactly to any of the response options given, but it's close to the third option, so there might be a typo or approximation error in your responses list. If the last option represents {−0.34, −11.66}, then there could be a typo in the equation or solution options provided. Please double-check the equation and the solutions.