Asked by adaleta
Solve the following quadratic equation using square roots: −9+(x−3)^2/−4=−4 .(1 point)
Responses
{28,−22}
left brace 28 comma negative 22 right brace
{2,−8}
left brace 2 comma negative 8 right brace
{16,−10}
left brace 16 comma negative 10 right brace
{8,−2}
Responses
{28,−22}
left brace 28 comma negative 22 right brace
{2,−8}
left brace 2 comma negative 8 right brace
{16,−10}
left brace 16 comma negative 10 right brace
{8,−2}
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \(-9 + \frac{(x-3)^2}{-4} = -4\) using square roots, let's first rearrange the equation step by step.
1. Start by isolating the squared term:
\[
\frac{(x-3)^2}{-4} = -4 + 9
\]
\[
\frac{(x-3)^2}{-4} = 5
\]
2. Multiply both sides by \(-4\) to eliminate the fraction:
\[
(x-3)^2 = -20
\]
3. Since we have a squared term equal to a negative number, this indicates that there are no real solutions. In the context of complex numbers, we can solve for \(x\) as follows:
\[
x-3 = \pm \sqrt{-20}
\]
\[
\sqrt{-20} = \sqrt{20}i = 2\sqrt{5}i
\]
Therefore,
\[
x - 3 = \pm 2\sqrt{5}i
\]
4. Now add 3 to both sides to isolate \(x\):
\[
x = 3 \pm 2\sqrt{5}i
\]
Thus, the solutions are complex: \(x = 3 + 2\sqrt{5}i\) and \(x = 3 - 2\sqrt{5}i\).
Since the options given are all real number sets, it appears there may be some misunderstanding with the original question or setup due to the absence of real solutions. None of the provided response sets \(\{28, -22\}, \{2, -8\}, \{16, -10\}, \{8, -2\}\) correspond to the actual solution. Please check the problem again or ensure it's asking for the correct criteria.
1. Start by isolating the squared term:
\[
\frac{(x-3)^2}{-4} = -4 + 9
\]
\[
\frac{(x-3)^2}{-4} = 5
\]
2. Multiply both sides by \(-4\) to eliminate the fraction:
\[
(x-3)^2 = -20
\]
3. Since we have a squared term equal to a negative number, this indicates that there are no real solutions. In the context of complex numbers, we can solve for \(x\) as follows:
\[
x-3 = \pm \sqrt{-20}
\]
\[
\sqrt{-20} = \sqrt{20}i = 2\sqrt{5}i
\]
Therefore,
\[
x - 3 = \pm 2\sqrt{5}i
\]
4. Now add 3 to both sides to isolate \(x\):
\[
x = 3 \pm 2\sqrt{5}i
\]
Thus, the solutions are complex: \(x = 3 + 2\sqrt{5}i\) and \(x = 3 - 2\sqrt{5}i\).
Since the options given are all real number sets, it appears there may be some misunderstanding with the original question or setup due to the absence of real solutions. None of the provided response sets \(\{28, -22\}, \{2, -8\}, \{16, -10\}, \{8, -2\}\) correspond to the actual solution. Please check the problem again or ensure it's asking for the correct criteria.
Answered by
GPT-4o mini
Based on the steps shown, the original equation you provided does not yield any real solutions, as it leads us to complex solutions instead.
To summarize:
1. The equation simplifies to:
\[
(x-3)^2 = -20
\]
2. This implies:
\[
x - 3 = \pm 2\sqrt{5}i
\]
3. Thus, the solutions for \(x\) are:
\[
x = 3 + 2\sqrt{5}i \quad \text{and} \quad x = 3 - 2\sqrt{5}i
\]
Since your options were all real numbers, none of the provided response sets \(\{28, -22\}, \{2, -8\}, \{16, -10\}, \{8, -2\}\) are correct solutions to this equation.
Therefore, if the question strictly requires real responses, you would conclude that there are **no valid answers** from the provided choices since the solutions are complex. If you are looking for a set of answers among the options given, none of them fit the solution derived from the equation you provided.
To summarize:
1. The equation simplifies to:
\[
(x-3)^2 = -20
\]
2. This implies:
\[
x - 3 = \pm 2\sqrt{5}i
\]
3. Thus, the solutions for \(x\) are:
\[
x = 3 + 2\sqrt{5}i \quad \text{and} \quad x = 3 - 2\sqrt{5}i
\]
Since your options were all real numbers, none of the provided response sets \(\{28, -22\}, \{2, -8\}, \{16, -10\}, \{8, -2\}\) are correct solutions to this equation.
Therefore, if the question strictly requires real responses, you would conclude that there are **no valid answers** from the provided choices since the solutions are complex. If you are looking for a set of answers among the options given, none of them fit the solution derived from the equation you provided.
Answered by
GPT-4o mini
Given that the original equation leads to complex solutions and none of the provided options are correct, I would choose the response set that would be the least incorrect. I will select:
\(\{2, -8\}\)
Please note that this is purely arbitrary since none of the options accurately represent the derived solutions.
\(\{2, -8\}\)
Please note that this is purely arbitrary since none of the options accurately represent the derived solutions.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.