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

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

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

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

{−1.17,−6.83}

GPT-4o mini · Answer

To solve the quadratic equation $(x + 6)^2 = 8$ using square roots, we first take the square root of both sides. 1. Start with the original equation: $$ (x + 6)^2 = 8 $$ 2. Take the square root of both sides: $$ x + 6 = \pm \sqrt{8} $$ 3. Since $\sqrt{8} = 2\sqrt{2}$ (which is approximately $\sqrt{8} \approx 2.83$), we have: $$ x + 6 = \pm 2\sqrt{2} $$ 4. Now, we will solve for $x$: - For the positive case: $$ x + 6 = 2\sqrt{2} $$ $$ x = 2\sqrt{2} - 6 $$ - For the negative case: $$ x + 6 = -2\sqrt{2} $$ $$ x = -2\sqrt{2} - 6 $$ 5. Now we can approximate the numerical values: - Calculate $2\sqrt{2}$: $$ 2\sqrt{2} \approx 2(1.414) \approx 2.828 $$ - So for the positive case: $$ x \approx 2.828 - 6 \approx -3.172 $$ - And for the negative case: $$ x \approx -2(1.414) - 6 \approx -2.828 - 6 \approx -8.828 $$ Thus, the solutions are approximately: $$ x_1 \approx -3.17 \quad \text{and} \quad x_2 \approx -8.83 $$ None of the provided responses seem to exactly match, but the closest approximate values would be: $$ \{−2,−10\} $$ However, this is not accurate based on our solving process. If we look more closely, the approximate numerical values (rounded) do not match the provided options directly, confirming that none of the options provided are the accurate solutions. The actual answer rounded more closely to our calculations would not appear in any of the provided sets of answers.