Question Use the discriminant to determine the number of real solutions to the quadratic equation. \[ -u^{2}+2 u-5=0 \]

To determine the number of real solutions to the quadratic equation \(-u^{2} + 2u - 5 = 0\), we first identify the coefficients: \(a = -1\), \(b = 2\), and \(c = -5\). The discriminant \(D\) is calculated using the formula \(D = b^{2} - 4ac\). Substituting the values, we get: \[D = (2)^{2} - 4(-1)(-5) = 4 - 20 = -16.\] Since the discriminant is negative (\(D < 0\)), this means there are no real solutions; instead, there are two complex solutions. If you're curious about the implications of complex solutions, they're often used in various fields such as engineering and physics, where wave behavior and oscillations misbehave in ways that real numbers can't fully capture! So, while you might not get real x-intercepts on a graph, you’re still venturing into the fascinating world of imaginary numbers, which open up a whole new dimension in problem-solving. And remember, next time you see a negative discriminant, rejoice! Because it means you're stepping into the mysterious realm of complex numbers, which can be just as exciting as any math class you’ve taken!