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. \( x+2=0 \) \( \pm 2 \pm i \sqrt{2} \pm 2 i \quad 2 \pm 4 i \sqrt{2} \quad-2 \pm 4 \sqrt{2} \pm \sqrt{2} \)

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: \(x=\pm \left(\sqrt{2}\times i\right)\) - step4: Separate into possible cases: \(\begin{align}&x=\sqrt{2}\times i\\&x=-\sqrt{2}\times i\end{align}\) - step5: Rewrite: \(x_{1}=-\sqrt{2}\times i,x_{2}=\sqrt{2}\times i\) 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: \(x=\pm \left(4\sqrt{2}\times i\right)\) - step3: Separate into possible cases: \(\begin{align}&x=4\sqrt{2}\times i\\&x=-4\sqrt{2}\times i\end{align}\) - step4: Rewrite: \(x_{1}=-4\sqrt{2}\times i,x_{2}=4\sqrt{2}\times i\) 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: - \( x^{2}=4 \) → \( \pm 2 \) - \( x^{2}=-4 \) → \( \pm 2i \) - \( x^{2}+2=0 \) → \( \pm \sqrt{2}i \) - \( x^{2}-2=0 \) → \( \pm \sqrt{2} \) - \( x^{2}=-32 \) → \( \pm 4\sqrt{2}i \) - \( x^{2}=32 \) → \( \pm 4\sqrt{2} \) - \( x-2=0 \) → \( 2 \) - \( x+2=0 \) → \( -2 \)