Final answer:
The discriminant for the equation -81a^2 + 90a - 25 = 0 is 0, indicating there is exactly one real solution to the equation.
Step-by-step explanation:
The student's question involves using the discriminant to determine the number of solutions to a quadratic equation. The quadratic equation in question is -81a^2 + 90a - 25 = 0. To find the discriminant, we use the formula Δ = b² - 4ac, where Δ is the discriminant and a, b, and c are coefficients from the quadratic equation ax² + bx + c = 0.
By substituting the coefficients from our equation we get Δ = (90)^2 - 4(-81)(-25). Calculating this gives us Δ = 8100 - 8100 = 0. Since the discriminant is zero, it means that there is exactly one real solution to the quadratic equation.