Question

What is the discriminant of `x^2-9=0` and what does that mean?

Answer 1

In your case, `Delta = 36`, which means that your equation has two distinct, rational solutions.

Explanation:

The general form of a quadratic equation is

`ax^2 + bx + c = 0`

for which the discriminant is defined as

`Delta = b^2 - 4 * a * c`

In your case, `a=1`, `b=0`, and `c=-9`, so the discriminant becomes

`Delta = 0^2 - 4 * 1 * (-9) = color(green)(36)`

A quadratic equation that has `Delta>0` has two distinct, real solutions. Moreover, since `Delta` is a perfect square, two two solutions will be rational numbers.

The general form for the solutions of a quadratic equation is

`color(blue)(x_(1,2) = (-b +- sqrt(Delta))/(2a)`

In your case, those two solutions will be

`x_(1,2) = (0 +- 6)/2 = {(x_1 = 3), (x_2 = -3) :}`

Note that these solutions could have easily been determined by

`x^2 - color(red)(cancel(color(black)(9))) + color(red)(cancel(color(black)(9))) = 9`

`sqrt(x^2) = sqrt(9) => x_(1,2) = +-3`