Question

What is an example of using the quadratic formula?

Answer 1

Suppose that you have a function represented by `f(x) = Ax^2 + Bx + C`.

We can use the quadratic formula to find the zeroes of this function, by setting `f(x) = Ax^2 + Bx + C = 0`.

Technically we can also find complex roots for it, but typically one will be asked to work only with real roots. The quadratic formula is represented as:

`(-B +- sqrt(B^2-4AC))/(2A) = x`

... where x represents the x-coordinate of the zero.

If `B^2 -4AC <0`, we will be dealing with complex roots, and if `B^2 - 4AC >=0`, we will have real roots.

As an example, consider the function `x^2 -13x + 12`. Here,

`A = 1, B = -13, C = 12.`

Then for the quadratic formula we would have:

`x = (13 +- sqrt ((-13)^2 - 4(1)(12)))/(2(1))` =

`(13 +- sqrt (169 - 48))/2 = (13+-11)/2`

Thus, our roots are `x=1` and `x=12`.

For an example with complex roots, we have the function `f(x) =x^2 +1`. Here `A = 1, B = 0, C = 1.`

Then by the quadratic equation,

`x = (0 +- sqrt (0^2 - 4(1)(1)))/(2(1)) = +-sqrt(-4)/2 = +-i`

... where `i` is the imaginary unit, defined by its property of `i^2 = -1`.

In the graph for this function on the real coordinate plane, we will see no zeroes, but the function will have these two imaginary roots.