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The discriminant determines the nature and number of solutions of a quadratic equation.
If b2 – 4ac is positive and greater than zero, there will be two distinct, real solutions. [2 real solutions appear from the magic discriminant ball]
If b2 – 4ac is zero, there will be one distinct, real solution. [1 real solution appears from the magic discriminant ball]
But, if b2 – 4ac is negative and less than zero, there are no real solutions.
Let's get quadratic for a couple of examples.
Our first example is x-squared plus three-x plus five equals zero. [The equation is written out]
Plugging these bad boys into the discriminant, b squared minus 4ac, we get three-squared
This simplifies to nine minus twenty, which is negative eleven.
This is less than zero…so there are no real solutions. [Guy holding his arms up]
What does “no real solutions” mean? [Students in class]
If we take a look at the graph, the parabola does not touch the x-axis at any x-value. [Finger going along the x-axis]
So it's not real, it doesn't exist.
For our second example, we have the equation...
x-squared minus six-x minus ten equals zero. [The equation is written out]
Plugging this into the discriminant it looks like...
six-squared minus four times negative-ten times one.
This is thirty-six plus forty, which is seventy-six.
Seventy-six is greater than zero.
So there are two distinct, real solutions to this problem. [Guy looks happy]
By looking at the graph, we can see that there are two real solutions because the parabola [Arrows point to the solutions on the graph]
touches the x-axis at two distinct x-values.
Okay, so the Magic Discriminant Ball might not be able to help you with some of life's [Guy asking the ball a random question]
biggest questions… [The magic discriminant ball looks angry]
… but using the discriminant can help you ace your next math test. [Guy looks happy as he gets an A+ on his test]