As every student has experienced some time in their school and university mathematics courses, the “general” quadratic equation is
and it can be solved by using the “quadratic formula”:
For the struggling student, this formula can be difficult to remember – all those coefficients! But it can easily be simplified. First, note that even though
is a general quadratic function, the general quadratic equation can in fact be simplified by dividing through by (which is assumed to be non-zero) to obtain
And in fact this step is the beginning of most derivations of the quadratic formula. Writing the coefficient as and the constant term as produces an equivalent general quadratic equation
for which the solution is
Isn’t that simpler?
A similar approach can be taken to the general “reduced cubic equation”
This is in fact completely general as any cubic equation can be put into this form by a linear transformation.
and cubing both sides produces:
Comparing coefficients with the cubic equation:
These can be easily solved to produce
This gives a solution to the cubic:
which is nearly simple enough to memorize. The other solutions are obtained by multiplying each term by various powers of , where and :