Question

How to use the discriminant to find out what type of solutions the equation has for `x^2 = 0`?

Answer 1

`x^2=0` is of the form `ax^2+bx+c = 0`, with `a=1`, `b=0` and `c=0`.

The discriminant is given by the formula:

`Delta = b^2-4ac = 0^2 - (4xx1xx0) = 0-0 = 0`

This means that `x^2=0` has one repeated real root and because `0=0^2` is a perfect square, that repeated root is rational.

The possible cases are:

`Delta < 0` No real roots. Two distinct complex roots.
`Delta = 0` One repeated, rational, real root.
`Delta > 0` Two distinct real roots. If `Delta` is also a perfect square then those roots are rational.

Severe overkill that it is, you can apply the standard quadratic solution formula to find the (repeated) solution as follows:

`x = (-b+-sqrt(Delta))/(2a) = (-0+-sqrt(0))/(2xx1) = 0/2 = 0`