Question Use the discriminant to determine the number of real solutions to the quadratic equation. \[ -4 z^{2}-3 z+5=0 \]

To determine the number of real solutions for the given quadratic equation, we can use the discriminant, which is given by \(D = b^2 - 4ac\). In your equation \(-4z^2 - 3z + 5 = 0\), we identify the coefficients as follows: - \(a = -4\) - \(b = -3\) - \(c = 5\) Now, plug these values into the discriminant formula: \[ D = (-3)^2 - 4(-4)(5) = 9 + 80 = 89 \] Since \(D > 0\), this means there are two distinct real solutions for the quadratic equation! Fun Fact: Quadratic equations can be visualized as parabolas. If the discriminant is positive, the parabola crosses the x-axis at two points, representing real solutions. If it’s zero, the vertex of the parabola just touches the x-axis (one real solution), and if it’s negative, the parabola floats above or below the x-axis without crossing it (no real solutions). Still curious? If you want to dive deeper into the world of quadratics, try playing with various examples and see how different values affect the discriminant and, in turn, the number of solutions! There are plenty of online quadratic calculators and apps that make exploring these concepts interactive and fun.