Simplify by using the imaginary unit \( i \) \( \frac{-10 \pm \sqrt{-48}}{2} \)

To simplify the expression \(\frac{-10 \pm \sqrt{-48}}{2}\) using the imaginary unit \(i\), follow these steps: 1. **Simplify the Square Root of a Negative Number:** \[ \sqrt{-48} = \sqrt{48 \times (-1)} = \sqrt{48} \times \sqrt{-1} = \sqrt{48} \times i \] 2. **Simplify \(\sqrt{48}\):** \[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \] So, \[ \sqrt{-48} = 4\sqrt{3} \, i \] 3. **Substitute Back into the Original Expression:** \[ \frac{-10 \pm \sqrt{-48}}{2} = \frac{-10 \pm 4\sqrt{3} \, i}{2} \] 4. **Simplify the Entire Expression by Dividing by 2:** \[ \frac{-10}{2} \pm \frac{4\sqrt{3} \, i}{2} = -5 \pm 2\sqrt{3} \, i \] **Final Answer:** \[ -5 \pm 2\sqrt{3}\,i \]