# How to determine the roots of a quadratic equation

1. Jan 29, 2006

### powp

Hello All

I am trying to figure out how to determine if there are two, one, zero, or infinitely many roots for given formula ax^2 + bx + c. Are there any easy ways to determine this with out having to use the quadratic formula?

Thanks

P

2. Jan 29, 2006

### Tide

I believe you are asking whether the polynomial has real roots. You can determine that by defining a function

$$f(x) = ax^2 + bx + c$$

and completing the square

$$f(x) = a \left(x + \frac {b}{2a}\right)^2 - \frac {b^2}{4a} + c$$

The graph of x is a parabola whose vertex is at

$$x = -\frac {b}{2a}$$

Given the sign of a you can determine whether the vertex is a maximum or a minimum and determine whether f(x) = 0 is possible.

3. Jan 29, 2006

### d_leet

Well if you're referring to roots in a general sense then yes there is an easy way, it's called the fundamental theorem of algebra. A polynomial of degree n has exactly n roots including multiplicities.

4. Jan 29, 2006

### 0rthodontist

Another way, maybe the way in your book, is using the discriminant b^2 - 4ac. If that's greater than 0 then the equation has 2 real roots, if it's less than 0 the equation has 0 real roots, and if it is equal to 0 then the equation has 1 real root. It comes out of the quadratic formula but you don't need to use the entire formula, just the part under the square root sign.