One thing that I learned this week in Pre Calc 11 is how to find the discriminant of a quadratic equation. The discriminant does not solve a quadratic equation, however it can help you avoid attempting to solve an equation with no solutions and it can also help determine the nature of the roots. The formula used to find the discriminant is also part of the quadratic fromula, There are three possible outcomes when you calculate the discriminant, it will either show you that the equation will have one solution, two solutions or no solutions at all. The discriminant is a really easy and fast way to check to determine whether or not it is worth taking the time to solve an entire equation.

If the discriminant is a number that is greater than zero then it will have two possible solutions and it also means that if you were to graph the equation it would intercept the axis two times. For example, if the discriminant is  than it is a distinct root, with two solutions, it is a rational root, because it is a perfect square and it is also a real root because it is greater than zero.

If the discriminant is a number that is equal to zero then it will have one possible solution and it will only intercept the axis one time on a graph. For example, if the discriminant is  than it is an equal root, with one solution, it is an irrational root and it is also a real root because it is greater than or equal to zero.

If the discriminant is a number that is less than zero then it will have no possible solutions and it will not intercept the axis at all. For example, if the discriminant is  than it is not a real root because it is less than zero (a negative number) and it will have no solutions.

I have included an example below that shows the detailed steps I would take to find the discriminant of a quadratic equation.