Question

What does it mean when zeros (solutions) are equal or unequal, and rational or irrational?

Answer 1

When zeros are equal, discriminant is `0`; when zeros are rational, discriminant is square of a rational number and when zeros are irrational, discriminant is not a square of a rational number.

Explanation:

I understand you are talking of quadratic polynomial like `f(x)=ax^2+bx+c`. The zeros of this function are given by

`(-b+-sqrt(b^2-4ac))/(2a)`

Here `b^2-4ac` is called the discriminant and its nature defines zeros of `f(x)`.

The `+-` signs give two roots to the polynomial, but if `b^2-4ac=0`, observe that we have only one solution `-b/(2a)`.

Further assume that `a,b` and `c` are rational. Now if discriminant is square of a rational number, the zeros will be rational

and if discriminant is not the square of a rational number, the zeros will be irrrational.