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

Certainly! Below is the matching of each equation with its corresponding solution(s): --- **1. \( x^{2} = 4 \)** - **Solutions:** \( x = 2 \) and \( x = -2 \) --- **2. \( x^{2} = -4 \)** - **Solutions:** \( x = 2i \) and \( x = -2i \) *(No real solutions; solutions are imaginary numbers.)* --- **3. \( x^{2} + 2 = 0 \)** - **Solutions:** \( x = i\sqrt{2} \) and \( x = -i\sqrt{2} \) *(No real solutions; solutions are imaginary numbers.)* --- **4. \( x^{2} - 2 = 0 \)** - **Solutions:** \( x = \sqrt{2} \) and \( x = -\sqrt{2} \) --- **5. \( x^{2} = -32 \)** - **Solutions:** \( x = 4\sqrt{2}\,i \) and \( x = -4\sqrt{2}\,i \) *(No real solutions; solutions are imaginary numbers.)* --- **6. \( x^{2} = 32 \)** - **Solutions:** \( x = 4\sqrt{2} \) and \( x = -4\sqrt{2} \) --- **7. \( x - 2 = 0 \)** - **Solution:** \( x = 2 \) --- **8. \( x + 2 = 0 \)** - **Solution:** \( x = -2 \) --- **Summary Table:** | Equation | Solutions | |------------------|------------------------------------| | \( x^{2} = 4 \) | \( x = 2, -2 \) | | \( x^{2} = -4 \) | \( x = 2i, -2i \) | | \( x^{2} + 2 = 0 \) | \( x = i\sqrt{2}, -i\sqrt{2} \) | | \( x^{2} - 2 = 0 \) | \( x = \sqrt{2}, -\sqrt{2} \) | | \( x^{2} = -32 \) | \( x = 4\sqrt{2}\,i, -4\sqrt{2}\,i \) | | \( x^{2} = 32 \) | \( x = 4\sqrt{2}, -4\sqrt{2} \) | | \( x - 2 = 0 \) | \( x = 2 \) | | \( x + 2 = 0 \) | \( x = -2 \) | --- **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!