The Discriminant
There are times when you don’t want to necessarily solve a quadratic function; you just want to figure out how many solutions there are. This may happen when you are analyzing a graph of a function.
In order to find out how many solutions a quadratic function has, you will use the Discriminant.
Let ax2 + bx + c = 0, where a ≠ 0.
The discriminant is found by using the formula b2 – 4ac.
If b2 – 4ac > 0, then the quadratic function has 2 distinct real solutions.
If b2 – 4ac = 0, then the quadratic function has 1 real solution, or a double root.
If b2 – 4ac < 0, then the quadratic function has 0 real solutions.
You may be asking yourself what it means for a quadratic to have no real solutions. The quadratic function still has two solutions, but they are imaginary. We will be discussing imaginary numbers in this section.
Let’s look at an example. Find the discriminant and the number of solutions for the equation:
2x2 – 4x + 1 = 0
The first step is to identify a, b, and c. Make sure that you keep in mind the negative signs. In this case:
a = 2, b = -4 and c = 1
Now plug these values into the discriminant.
Since the discriminant is greater than zero, there are two distinct real solutions to this quadratic equation.b2 – 4ac = (-4)2 – 4(2)(1) = 16 – 8 = 8
