Match each equation below with its solution(s) $$ \begin{array}{ll} ext { 1. } x^{2}=4 & ext { 2. } x^{2}=-4 \ \begin{array}{ll} ext { 3. } x^{2}+2=0 & ext { 4. } x^{2}-2=0 \ ext { 5. } x^{2}=-32 & ext { 6. } x^{2}=32 \ ext { 7. } x-2=0 & ext { 8. } x+2=0\end{array}\end{array} $$

Question

Match each equation below with its solution(s) $$ \begin{array}{ll}	ext { 1. } x^{2}=4 & 	ext { 2. } x^{2}=-4 \ \begin{array}{ll}	ext { 3. } x^{2}+2=0 & 	ext { 4. } x^{2}-2=0 \ 	ext { 5. } x^{2}=-32 & 	ext { 6. } x^{2}=32 \ 	ext { 7. } x-2=0 & 	ext { 8. } x+2=0\end{array}\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Below is the matching of each equation with its corresponding solution(s):

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**1. $$ x^{2} = 4 $$**

- **Solutions:** $$ x = 2 $$ and $$ x = -2 $$

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**2. $$ x^{2} = -4 $$**

- **Solutions:** $$ x = 2i $$ and $$ x = -2i $$  
  *(No real solutions; solutions are imaginary numbers.)*

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**3. $$ x^{2} + 2 = 0 $$**

- **Solutions:** $$ x = i\sqrt{2} $$ and $$ x = -i\sqrt{2} $$  
  *(No real solutions; solutions are imaginary numbers.)*

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**4. $$ x^{2} - 2 = 0 $$**

- **Solutions:** $$ x = \sqrt{2} $$ and $$ x = -\sqrt{2} $$

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**5. $$ x^{2} = -32 $$**

- **Solutions:** $$ x = 4\sqrt{2}\,i $$ and $$ x = -4\sqrt{2}\,i $$  
  *(No real solutions; solutions are imaginary numbers.)*

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**6. $$ x^{2} = 32 $$**

- **Solutions:** $$ x = 4\sqrt{2} $$ and $$ x = -4\sqrt{2} $$

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**7. $$ x - 2 = 0 $$**

- **Solution:** $$ x = 2 $$

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**8. $$ x + 2 = 0 $$**

- **Solution:** $$ x = -2 $$

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**Summary Table:**

| Equation         | Solutions                          |
|------------------|------------------------------------|
| $$ x^{2} = 4 $$      | $$ x = 2, -2 $$                      |
| $$ x^{2} = -4 $$     | $$ x = 2i, -2i $$                    |
| $$ x^{2} + 2 = 0 $$  | $$ x = i\sqrt{2}, -i\sqrt{2} $$      |
| $$ x^{2} - 2 = 0 $$  | $$ x = \sqrt{2}, -\sqrt{2} $$        |
| $$ x^{2} = -32 $$    | $$ x = 4\sqrt{2}\,i, -4\sqrt{2}\,i $$ |
| $$ x^{2} = 32 $$     | $$ x = 4\sqrt{2}, -4\sqrt{2} $$      |
| $$ x - 2 = 0 $$      | $$ x = 2 $$                           |
| $$ x + 2 = 0 $$      | $$ x = -2 $$                          |

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**Notes:**

- **Real Solutions:** Equations 1, 4, 6, 7, and 8 have real solutions.
- **Imaginary Solutions:** Equations 2, 3, and 5 have no real solutions and instead have imaginary (complex) solutions.

If you have any further questions or need additional explanations, feel free to ask!
Here are the solutions for each equation: 1. $$ x^{2} = 4 $$ → $$ x = 2 $$ and $$ x = -2 $$ 2. $$ x^{2} = -4 $$ → $$ x = 2i $$ and $$ x = -2i $$ 3. $$ x^{2} + 2 = 0 $$ → $$ x = i\sqrt{2} $$ and $$ x = -i\sqrt{2} $$ 4. $$ x^{2} - 2 = 0 $$ → $$ x = \sqrt{2} $$ and $$ x = -\sqrt{2} $$ 5. $$ x^{2} = -32 $$ → $$ x = 4\sqrt{2}\,i $$ and $$ x = -4\sqrt{2}\,i $$ 6. $$ x^{2} = 32 $$ → $$ x = 4\sqrt{2} $$ and $$ x = -4\sqrt{2} $$ 7. $$ x - 2 = 0 $$ → $$ x = 2 $$ 8. $$ x + 2 = 0 $$ → $$ x = -2 $$

Match each equation below with its solution(s)

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