The Quadratic Formula

In algebra , one frequently wishes to find those values of x for which an equation of the
form ax² + bx + c = 0 holds. Such an equation is called a Quadratic Equation , and its two
solutions can be found using the Quadratic Formula:


This formula at first might seem mysterious, but we can take away the mystery by showing
how to derive it. We’ll start with a few simple cases , and work our way up to the most
general case.

First, it’s easy to solve an equation like x ² = s, where s is just some real number . We
just need x to be one of the two numbers whose square is equal to s, so we want x =
If s is negative , this leads to imaginary solutions, but we won’t worry about this.

It’s not much more difficult to solve an equation like (x + r)² = s. In this case, we need
x + r to be So, we need x to be −r ± Note that the equation (x + r)² = s
could be rewritten as x² + 2rx + r² = s, or as x² + 2rx + r² − s = 0. This form of
the equation is beginning to look a little more like the general quadratic equation that we
started out with. We can make it look even more like that equation by multiplying the entire
equation by a non zero number a. Remember that multiplying both sides of an equation by
a nonzero number doesn ’t change the solutions of the equation! So, the two solutions of
ax² + 2rax + (r² − s)a = 0 are x = −r ±

Now let’s consider the general quadratic equation: ax² +bx+c = 0. We can rewrite this
equation in the form ax² +2rax+(r² −s)a = 0 by choosing the right values for r and s. In
particular, we want 2ra to be equal to b (so r = b/2a ), and we want (r² − s)a to be equal to
c. That means that r² − s = c/a, so s = r² − (c/a). Since we already decided that r = b/2a ,

we need s to equal We can simplify this to


So, with these values for r and s, the equation ax² + bx + c = 0 has to have the same
solutions as the equation ax² + 2rax + (r² − s)a = 0, since these are the same equations.
We’ve already found those solutions to be x = −r±So now, all we have to do is replace
r and s by their equivalent values , and we get two solutions for x:


Notice thatWe need to be careful here, because That is,
when a is itself negative,
Therefore, is either
equal to or it’s equal to Still, the two values
are the same as the two values So we can write the two solutions in the form

That is the quadratic formula!