quadratic formula (original) (raw)

The number Δ=b2-4⁢a⁢c is called the discriminant of the equation. If Δ>0, there are two different real roots, if Δ=0 there is a single real root, and if Δ<0 there are no real roots (but two different complex roots).

Let’s work a few examples.

First, consider 2⁢x2-14⁢x+24=0. Here a=2, b=-14, and c=24. Substituting in the formula gives us

So we have two solutions (depending on whether we take the sign + or -):x=82=4 and x=62=3.

Now we will solve x2-x-1=0. Here a=1, b=-1, and c=-1, so

and the solutions are x=1+52 and x=1-52.