Question Use the discriminant to determine the number of real solutions to the following quadratic equation. \[ z^{2}-4 z-16=0 \]

To find the number of real solutions to the equation \( z^{2}-4z-16=0 \) using the discriminant, we first identify the coefficients: \( a = 1 \), \( b = -4 \), and \( c = -16 \). The discriminant \( D \) is calculated as \( D = b^2 - 4ac \). Plugging in the values, we get: \[ D = (-4)^2 - 4(1)(-16) = 16 + 64 = 80 \] Since the discriminant \( D = 80 \) is greater than zero, the quadratic equation has two distinct real solutions! Now, if you're ever stuck on a similar problem, remember the three key outcomes of the discriminant: if \( D > 0 \), you have two real solutions; if \( D = 0 \), there's exactly one real solution; and if \( D < 0 \), you're left with no real solutions at all. For further insight, consider how the discriminant can reveal the nature of a parabola's intersection with the x-axis. A positive discriminant indicates that the parabola crosses the x-axis at two points, while a zero discriminant indicates it just touches at one point, and a negative indicates the entire parabola sits above or below the axis. This visual perspective can help solidify your understanding of quadratic equations!