Quadratic Solution B7347B

1. **State the problem:** Solve the quadratic equation $$z^2 + 10z + 15 = 0$$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$az^2 + bz + c = 0$$, the solutions are given by: $$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 1$$, $$b = 10$$, and $$c = 15$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 10^2 - 4 \times 1 \times 15 = 100 - 60 = 40$$ 5. **Apply the quadratic formula:** $$z = \frac{-10 \pm \sqrt{40}}{2 \times 1} = \frac{-10 \pm \sqrt{40}}{2}$$ 6. **Simplify the square root:** $$\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$$ 7. **Substitute back:** $$z = \frac{-10 \pm 2\sqrt{10}}{2}$$ 8. **Simplify the fraction by canceling 2:** $$z = \frac{\cancel{2}(-5 \pm \sqrt{10})}{\cancel{2}} = -5 \pm \sqrt{10}$$ **Final answer:** $$z = -5 \pm \sqrt{10}$$ which corresponds to option D.