Match each equation below with its solution(s) 1. $$ x^{2}=4 $$ 2. $$ x^{2}=-4 $$ 3. $$ x^{2}+2=0 $$ 4. $$ x^{2}-2=0 $$ 5. $$ x^{2}=-32 $$ 6. $$ x^{2}=32 $$ 7. $$ x-2=0 $$ 8. $$ x+2=0 $$ $$ \pm 2 \pm i \sqrt{2} \pm 2 i \quad 2 \pm 4 i \sqrt{2} \quad-2 \pm 4 \sqrt{2} \pm \sqrt{2} $$

Question

UpStudy AI · Accepted Answer

Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x-2=0$$
- step1: Move the constant to the right side:
  $$x=0+2$$
- step2: Remove 0:
  $$x=2$$
Solve the equation $$ x+2=0 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x+2=0$$
- step1: Move the constant to the right side:
  $$x=0-2$$
- step2: Remove 0:
  $$x=-2$$
Solve the equation $$ x^{2}=4 $$.
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
  $$x^{2}=4$$
- step1: Simplify the expression:
  $$x=\pm \sqrt{4}$$
- step2: Simplify the expression:
  $$x=\pm 2$$
- step3: Separate into possible cases:
  $$\begin{align}&x=2\&x=-2\end{align}$$
- step4: Rewrite:
  $$x_{1}=-2,x_{2}=2$$
Solve the equation $$ x^{2}=-4 $$.
Solve the equation(The complex numbers system) by following steps:
- step0: Solve using square roots:
  $$x^{2}=-4$$
- step1: Simplify the expression:
  $$x=\pm \sqrt{-4}$$
- step2: Simplify the expression:
  $$x=\pm 2i$$
- step3: Separate into possible cases:
  $$\begin{align}&x=2i\&x=-2i\end{align}$$
- step4: Rewrite:
  $$x_{1}=-2i,x_{2}=2i$$
Solve the equation $$ x^{2}-2=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
  $$x^{2}-2=0$$
- step1: Add or subtract both sides:
  $$x^{2}=2$$
- step2: Simplify the expression:
  $$x=\pm \sqrt{2}$$
- step3: Separate into possible cases:
  $$\begin{align}&x=\sqrt{2}\&x=-\sqrt{2}\end{align}$$
- step4: Rewrite:
  $$x_{1}=-\sqrt{2},x_{2}=\sqrt{2}$$
Solve the equation $$ x^{2}+2=0 $$.
Solve the equation(The complex numbers system) by following steps:
- step0: Solve using square roots:
  $$x^{2}+2=0$$
- step1: Add or subtract both sides:
  $$x^{2}=-2$$
- step2: Simplify the expression:
  $$x=\pm \sqrt{-2}$$
- step3: Simplify the expression:
  $$x=\pm \left(\sqrt{2}	imes iight)$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\sqrt{2}	imes i\&x=-\sqrt{2}	imes i\end{align}$$
- step5: Rewrite:
  $$x_{1}=-\sqrt{2}	imes i,x_{2}=\sqrt{2}	imes i$$
Solve the equation $$ x^{2}=32 $$.
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
  $$x^{2}=32$$
- step1: Simplify the expression:
  $$x=\pm \sqrt{32}$$
- step2: Simplify the expression:
  $$x=\pm 4\sqrt{2}$$
- step3: Separate into possible cases:
  $$\begin{align}&x=4\sqrt{2}\&x=-4\sqrt{2}\end{align}$$
- step4: Rewrite:
  $$x_{1}=-4\sqrt{2},x_{2}=4\sqrt{2}$$
Solve the equation $$ x^{2}=-32 $$.
Solve the equation(The complex numbers system) by following steps:
- step0: Solve using square roots:
  $$x^{2}=-32$$
- step1: Simplify the expression:
  $$x=\pm \sqrt{-32}$$
- step2: Simplify the expression:
  $$x=\pm \left(4\sqrt{2}	imes iight)$$
- step3: Separate into possible cases:
  $$\begin{align}&x=4\sqrt{2}	imes i\&x=-4\sqrt{2}	imes i\end{align}$$
- step4: Rewrite:
  $$x_{1}=-4\sqrt{2}	imes i,x_{2}=4\sqrt{2}	imes i$$
Let's match each equation with its corresponding solution(s):

1. **Equation:** $$ x^{2}=4 $$  
   **Solution:** $$ x_{1}=-2, x_{2}=2 $$  
   **Match:** $$ \pm 2 $$

2. **Equation:** $$ x^{2}=-4 $$  
   **Solution:** $$ x_{1}=-2i, x_{2}=2i $$  
   **Match:** $$ \pm 2i $$

3. **Equation:** $$ x^{2}+2=0 $$  
   **Solution:** $$ x_{1}=-\sqrt{2}i, x_{2}=\sqrt{2}i $$  
   **Match:** $$ \pm \sqrt{2}i $$

4. **Equation:** $$ x^{2}-2=0 $$  
   **Solution:** $$ x_{1}=-\sqrt{2}, x_{2}=\sqrt{2} $$  
   **Match:** $$ \pm \sqrt{2} $$

5. **Equation:** $$ x^{2}=-32 $$  
   **Solution:** $$ x_{1}=-4\sqrt{2}i, x_{2}=4\sqrt{2}i $$  
   **Match:** $$ \pm 4\sqrt{2}i $$

6. **Equation:** $$ x^{2}=32 $$  
   **Solution:** $$ x_{1}=-4\sqrt{2}, x_{2}=4\sqrt{2} $$  
   **Match:** $$ \pm 4\sqrt{2} $$

7. **Equation:** $$ x-2=0 $$  
   **Solution:** $$ x=2 $$  
   **Match:** $$ 2 $$

8. **Equation:** $$ x+2=0 $$  
   **Solution:** $$ x=-2 $$  
   **Match:** $$ -2 $$

### Summary of Matches:
- $$ x^{2}=4 $$ → $$ \pm 2 $$
- $$ x^{2}=-4 $$ → $$ \pm 2i $$
- $$ x^{2}+2=0 $$ → $$ \pm \sqrt{2}i $$
- $$ x^{2}-2=0 $$ → $$ \pm \sqrt{2} $$
- $$ x^{2}=-32 $$ → $$ \pm 4\sqrt{2}i $$
- $$ x^{2}=32 $$ → $$ \pm 4\sqrt{2} $$
- $$ x-2=0 $$ → $$ 2 $$
- $$ x+2=0 $$ → $$ -2 $$
Here are the matches between each equation and its solution(s): 1. $$ x^{2}=4 $$ → $$ \pm 2 $$ 2. $$ x^{2}=-4 $$ → $$ \pm 2i $$ 3. $$ x^{2}+2=0 $$ → $$ \pm \sqrt{2}i $$ 4. $$ x^{2}-2=0 $$ → $$ \pm \sqrt{2} $$ 5. $$ x^{2}=-32 $$ → $$ \pm 4\sqrt{2}i $$ 6. $$ x^{2}=32 $$ → $$ \pm 4\sqrt{2} $$ 7. $$ x-2=0 $$ → $$ 2 $$ 8. $$ x+2=0 $$ → $$ -2 $$

Match each equation below with its solution(s) 1. \( x^{2}=4 \) 2. \( x^{2}=-4 \) 3. \( x^{2}+2=0 \) 4. \( x^{2}-2=0 \) 5. \( x^{2}=-32 \) 6. \( x^{2}=32 \) 7. \( x-2=0 \) 8. \( x+2=0 \) \( \pm 2 \pm i \sqrt{2} \pm 2 i \quad 2 \pm 4 i \sqrt{2} \quad-2 \pm 4 \sqrt{2} \pm \sqrt{2} \)

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