Question

How to use the discriminant to find out how many real number roots an equation has for `x^2+4x+5`?

Answer 1

`x^2+4x+5` is of the form `ax^2+bx+c` with `a=1`, `b=4` and `c=5`.

The discriminant is given by the formula:

`Delta = b^2-4ac = 4^2-(4xx1xx5) = 16-20 = -4`

Since this is negative, `x^2+4x+5=0` has no real roots. It has two distinct complex roots.

The various possible cases are:
`Delta > 0` : The quadratic has two distinct real roots.
`Delta = 0` : The quadratic has one repeated real root.
`Delta < 0` : The quadratic has no real roots. It has two distinct complex roots.

In addition, if the original coefficients are integers (or rational numbers) and `Delta >= 0` is a perfect square, then the roots are rational numbers.