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

Certainly! Let's match each equation with its corresponding solution(s). ### Equations and Their Solutions: 1. **\( x^{2} = 4 \)** - **Solutions:** \( x = \pm 2 \) 2. **\( x^{2} = -4 \)** - **Solutions:** \( x = \pm 2i \) 3. **\( x^{2} + 2 = 0 \)** - **Solutions:** \( x = \pm i\sqrt{2} \) 4. **\( x^{2} - 2 = 0 \)** - **Solutions:** \( x = \pm \sqrt{2} \) 5. **\( x^{2} = -32 \)** - **Solutions:** \( x = \pm 4\sqrt{2}\,i \) 6. **\( x^{2} = 32 \)** - **Solutions:** \( x = \pm 4\sqrt{2} \) 7. **\( x - 2 = 0 \)** - **Solution:** \( x = 2 \) 8. **\( x + 2 = 0 \)** - **Solution:** \( x = -2 \) ### Summary of Matches: | **Equation** | **Solution(s)** | |-------------------------|---------------------------| | \( 1. \, x^{2} = 4 \) | \( \pm 2 \) | | \( 2. \, x^{2} = -4 \) | \( \pm 2i \) | | \( 3. \, x^{2} + 2 = 0 \) | \( \pm i\sqrt{2} \) | | \( 4. \, x^{2} - 2 = 0 \) | \( \pm \sqrt{2} \) | | \( 5. \, x^{2} = -32 \) | \( \pm 4\sqrt{2}\,i \) | | \( 6. \, x^{2} = 32 \) | \( \pm 4\sqrt{2} \) | | \( 7. \, x - 2 = 0 \) | \( 2 \) | | \( 8. \, x + 2 = 0 \) | \( -2 \) | ### Explanation: - **Quadratic Equations (\(x^2 = \text{constant}\)):** - **Positive Constant:** The solutions are real numbers. - **Negative Constant:** The solutions involve imaginary numbers (\(i\)). - **Linear Equations (\(x \pm \text{constant} = 0\)):** - These have straightforward real solutions. Feel free to reach out if you need further clarification on any of these solutions!