This expression is called the Discriminant, also shown as Δ. It is equal to b² - 4ac. This is a very important part of the quadratic formula as it determines whether x will have two values, one repeated value or no real values. Here are a few examples.
a) x² - 2x - 1. a is equal to 1 since 1x² = x². b = -2, c = -1 The discriminant will be (-2)² - 4×1×-1 = 4 + 4 = 8. Since Δ > 0, there are two x values. Graphed, the parabola sinks below the x axis.
b) x². a = 1, b = 0 (0x = 0), c = 0 The discriminant will be 0² - 4×1×0 = 0 - 0 = 0. Since Δ = 0, there is only one x value. Graphed, the parabola touches the x axis at only one point.
c) x² + 1. a = 1, c = 1. The discriminant will be 0² - 4×1×1 = 0 - 4 = -4 Since Δ < 0, there are no real x values. Graphed, the parabola floats above the x axis.
