Question

What is the discriminant of a quadratic and why does it tell us the number of solutions?

Answer 1

See below:

Explanation:

I'd be glad to offer a little "primer" on discriminants of quadratics:

If we have the quadratic

`ax^2+bx+c`, we can think about how many solutions it as by analyzing the discriminant

`bar( ul|color(white)(2/2)b^2-4ac color(white)(2/2)|)`

We have three cases:

`b^2-4ac>0=>2` real solutins

`b^2-4ac<0=>` no real solutions

`b^2-4ac=0=>1` real solution

Why does this makes sense?

Recall the Quadratic Formula

`x=(-b+-sqrt(color(blue)(b^2-4ac)))/(2a)`

What do you notice? The discriminant is the value under the radical.

If we want to think about how many solutions a quadratic has, the `-b` and `2a` won't affect that...what will is our value under the radical, which we call, the discriminant.

Remember that if we take the square root of a negative number, we get imaginary values. So if `b^2-4ac<0`, we have no real solutions.

What happens when `b^2-4ac>0`? We're dealing with positive numbers, which have two values- a positive and a negative solution.

When `b^2-4ac=0`, we are just taking the square root of zero, which is zero and only zero. Thus, we will have one solution.

Whatever is under the radical determines how many solutions we'll have.

I want to clarify that to determine the number of solutions, you don't have to plug the values into the Quadratic Formula. Plug in the values into the discriminant, and see what you get.

Hope this helps!