Solve the quadratic equation. \( -2 z^{2}-2 z+5=0 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution(s) is(are) \( z= \) (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. There are no real solutions.

To solve the quadratic equation \(-2z^2 - 2z + 5 = 0\), we can start by using the quadratic formula \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = -2\), \(b = -2\), and \(c = 5\). First, we calculate the discriminant: \[ b^2 - 4ac = (-2)^2 - 4(-2)(5) = 4 + 40 = 44. \] Since the discriminant is positive, we can find two real solutions. Now, substituting into the quadratic formula: \[ z = \frac{-(-2) \pm \sqrt{44}}{2(-2)} = \frac{2 \pm 2\sqrt{11}}{-4} = -\frac{1 \pm \sqrt{11}}{2}. \] Thus, the solutions are: \[ z = -\frac{1 + \sqrt{11}}{2}, -\frac{1 - \sqrt{11}}{2}. \] So, the correct choice is A. The solution(s) is(are) \( z = -\frac{1 + \sqrt{11}}{2}, -\frac{1 - \sqrt{11}}{2} \). The quadratic formula, first introduced by ancient Greeks, becomes even more interesting when we trace back to how different cultures approached it. The Babylonians had geometric methods to solve these equations long before algebra was formalized, using sophisticated techniques without the notation we use today! In practical applications, quadratic equations are everywhere! From physics problems involving projectile motion to economics in optimizing profit functions, understanding how to solve them helps not just in academics but in deciphering real-world phenomena and making informed decisions.