Question
Solve the following quadratic equation using square roots: −9+(x−3)^2/−4=−4
.(1 point)
{2,−8}
{8,−2}
{16,−10}
{28,−22}
.(1 point)
{2,−8}
{8,−2}
{16,−10}
{28,−22}
Answers
Answer
Solve the following quadratic equation using square roots: −9+(x−3)^2/(−4)=−4
.(1 point)
{2,−8}
{8,−2}
{16,−10}
{28,−22}
.(1 point)
{2,−8}
{8,−2}
{16,−10}
{28,−22}
Answer
ok now this time divide (x - 3)^2 by -4
Answered by
GPT-4o mini
To solve the equation \(-9 + \frac{(x - 3)^2}{-4} = -4\), let's first simplify it step-by-step.
1. **Isolate the square term**:
\[
-9 + \frac{(x - 3)^2}{-4} = -4
\]
Adding 9 to both sides:
\[
\frac{(x - 3)^2}{-4} = 5
\]
2. **Multiply both sides by \(-4\)** to get rid of the fraction:
\[
(x - 3)^2 = -20
\]
3. **Recognize that \(-20\) indicates a complex solution**, since the square of a real number cannot be negative. To proceed to find the square roots:
\[
x - 3 = \pm \sqrt{-20}
\]
This simplifies to:
\[
x - 3 = \pm \sqrt{20}i = \pm 2\sqrt{5}i
\]
4. **Solve for \(x\)**:
\[
x = 3 \pm 2\sqrt{5}i
\]
Since the problem asks for solutions in real numbers, we conclude that the quadratic equation has no real solutions. Therefore, none of the provided answer sets \(\{2,−8\}\), \(\{8,−2\}\), \(\{16,−10\}\), or \(\{28,−22\}\) are correct.
The solutions are \(3 + 2\sqrt{5}i\) and \(3 - 2\sqrt{5}i\) if complex numbers are acceptable from your context. Let me know if you need further assistance!
1. **Isolate the square term**:
\[
-9 + \frac{(x - 3)^2}{-4} = -4
\]
Adding 9 to both sides:
\[
\frac{(x - 3)^2}{-4} = 5
\]
2. **Multiply both sides by \(-4\)** to get rid of the fraction:
\[
(x - 3)^2 = -20
\]
3. **Recognize that \(-20\) indicates a complex solution**, since the square of a real number cannot be negative. To proceed to find the square roots:
\[
x - 3 = \pm \sqrt{-20}
\]
This simplifies to:
\[
x - 3 = \pm \sqrt{20}i = \pm 2\sqrt{5}i
\]
4. **Solve for \(x\)**:
\[
x = 3 \pm 2\sqrt{5}i
\]
Since the problem asks for solutions in real numbers, we conclude that the quadratic equation has no real solutions. Therefore, none of the provided answer sets \(\{2,−8\}\), \(\{8,−2\}\), \(\{16,−10\}\), or \(\{28,−22\}\) are correct.
The solutions are \(3 + 2\sqrt{5}i\) and \(3 - 2\sqrt{5}i\) if complex numbers are acceptable from your context. Let me know if you need further assistance!
Answered by
GPT-4o mini
To solve the quadratic equation \( -9 + \frac{(x - 3)^2}{-4} = -4 \), let's isolate \((x - 3)^2\) step by step.
1. Start with the original equation:
\[
-9 + \frac{(x - 3)^2}{-4} = -4
\]
2. Add \(9\) to both sides to simplify:
\[
\frac{(x - 3)^2}{-4} = -4 + 9
\]
\[
\frac{(x - 3)^2}{-4} = 5
\]
3. Multiply both sides by \(-4\) to eliminate the fraction:
\[
(x - 3)^2 = -20
\]
4. Notice that we have a negative value on the right side, which indicates there are no real solutions because you cannot take the square root of a negative number in the real number system. However, if sticking to the real solutions is required, we can write the following:
Because \((x - 3)^2 = -20\) leads to complex solutions:
\[
x - 3 = \pm \sqrt{-20}
\]
\[
x - 3 = \pm 2i\sqrt{5}
\]
\[
x = 3 \pm 2i\sqrt{5}
\]
Yet, since you are requesting a specific numerical solution from your provided options:
- **None of those options** correspond to the above finding or are valid answers for the equation provided.
If we consider additional context or the nature of quadratic equations, please clarify further for us to find the most suitable calculation that leads to an appropriate answer set like \{2, -8\}, \{8, -2\}, \{16, -10\}, or \{28, -22\}, as none of those options align with the calculations presented above.
If the equation \(-9 + \frac{(x-3)^2}{-4} = -4\) is used correctly, understanding the domain of solutions (real vs complex) becomes crucial in applying the right output. Please let me know how else I can assist you!
1. Start with the original equation:
\[
-9 + \frac{(x - 3)^2}{-4} = -4
\]
2. Add \(9\) to both sides to simplify:
\[
\frac{(x - 3)^2}{-4} = -4 + 9
\]
\[
\frac{(x - 3)^2}{-4} = 5
\]
3. Multiply both sides by \(-4\) to eliminate the fraction:
\[
(x - 3)^2 = -20
\]
4. Notice that we have a negative value on the right side, which indicates there are no real solutions because you cannot take the square root of a negative number in the real number system. However, if sticking to the real solutions is required, we can write the following:
Because \((x - 3)^2 = -20\) leads to complex solutions:
\[
x - 3 = \pm \sqrt{-20}
\]
\[
x - 3 = \pm 2i\sqrt{5}
\]
\[
x = 3 \pm 2i\sqrt{5}
\]
Yet, since you are requesting a specific numerical solution from your provided options:
- **None of those options** correspond to the above finding or are valid answers for the equation provided.
If we consider additional context or the nature of quadratic equations, please clarify further for us to find the most suitable calculation that leads to an appropriate answer set like \{2, -8\}, \{8, -2\}, \{16, -10\}, or \{28, -22\}, as none of those options align with the calculations presented above.
If the equation \(-9 + \frac{(x-3)^2}{-4} = -4\) is used correctly, understanding the domain of solutions (real vs complex) becomes crucial in applying the right output. Please let me know how else I can assist you!
Answered by
GPT-4o mini
Let's follow the steps for the equation you provided, focusing specifically on the operation of dividing \((x - 3)^2\) by \(-4\) correctly.
The given equation is:
\[
-9 + \frac{(x - 3)^2}{-4} = -4
\]
Now, we will first isolate \(\frac{(x - 3)^2}{-4}\) by adding \(9\) to both sides:
1. **Isolate the fraction**:
\[
\frac{(x - 3)^2}{-4} = -4 + 9
\]
\[
\frac{(x - 3)^2}{-4} = 5
\]
2. **Multiply both sides by -4** to eliminate the denominator:
\[
(x - 3)^2 = 5 \cdot -4
\]
\[
(x - 3)^2 = -20
\]
Now, we realize that this leads us to a negative square. Let's address solving this step properly:
3. Take the square root of both sides:
\[
x - 3 = \pm \sqrt{-20}
\]
4. We will convert the negative under the square root:
\[
x - 3 = \pm \sqrt{20} i = \pm 2\sqrt{5} i
\]
5. Finally, solve for \(x\):
\[
x = 3 \pm 2\sqrt{5} i
\]
As we derived previously, we have two complex solutions here.
However, if you want to double-check how to divide a square term by \(-4\):
When dealing only with \((x - 3)^2\) divided by \(-4\):
- Dividing \((x - 3)^2\) results in \(-\frac{(x - 3)^2}{4}\).
If there are real integer solutions or a misunderstanding in the function required to check, while resolving real values only or linking conditions to re-write to quadratic form, please clarify for the correct numerical resolution.
The given equation is:
\[
-9 + \frac{(x - 3)^2}{-4} = -4
\]
Now, we will first isolate \(\frac{(x - 3)^2}{-4}\) by adding \(9\) to both sides:
1. **Isolate the fraction**:
\[
\frac{(x - 3)^2}{-4} = -4 + 9
\]
\[
\frac{(x - 3)^2}{-4} = 5
\]
2. **Multiply both sides by -4** to eliminate the denominator:
\[
(x - 3)^2 = 5 \cdot -4
\]
\[
(x - 3)^2 = -20
\]
Now, we realize that this leads us to a negative square. Let's address solving this step properly:
3. Take the square root of both sides:
\[
x - 3 = \pm \sqrt{-20}
\]
4. We will convert the negative under the square root:
\[
x - 3 = \pm \sqrt{20} i = \pm 2\sqrt{5} i
\]
5. Finally, solve for \(x\):
\[
x = 3 \pm 2\sqrt{5} i
\]
As we derived previously, we have two complex solutions here.
However, if you want to double-check how to divide a square term by \(-4\):
When dealing only with \((x - 3)^2\) divided by \(-4\):
- Dividing \((x - 3)^2\) results in \(-\frac{(x - 3)^2}{4}\).
If there are real integer solutions or a misunderstanding in the function required to check, while resolving real values only or linking conditions to re-write to quadratic form, please clarify for the correct numerical resolution.
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