The quadratic formula

As every student has experienced some time in their school and university mathematics courses, the “general” quadratic equation is

ax^2+bx+c=0

and it can be solved by using the “quadratic formula”:

\displaystyle{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}.

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

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

is a general quadratic function, the general quadratic equation can in fact be simplified by dividing through by a (which is assumed to be non-zero) to obtain

\displaystyle{x^2+\frac{b}{a}x+\frac{c}{a}=0}.

And in fact this step is the beginning of most derivations of the quadratic formula. Writing the x coefficient as 2a and the constant term as b produces an equivalent general quadratic equation

x^2+2ax+b=0

for which the solution is

x=-a\pm\sqrt{a^2-b}.

Isn’t that simpler?

A similar approach can be taken to the general “reduced cubic equation”

x^3+3ax+2b=0.

This is in fact completely general as any cubic equation can be put into this form by a linear transformation.

Writing

x=p^{1/3}+q^{1/3}

and cubing both sides produces:

x^3=p+q+3p^{1/3}q^{1/3}x.

Comparing coefficients with the cubic equation:

a = -p^{1/3}q^{1/3}

or

pq=-a^3

and

p+q=-2b

These can be easily solved to produce

\displaystyle{p,q=-b\pm\sqrt{b^2+a^3}}.

This gives a solution to the cubic:

x=\displaystyle{\left(-b+\sqrt{b^2+a^3}\right)^{\!1/3}+\left(-b-\sqrt{b^2+a^3}\right)^{\!1/3}}

which is nearly simple enough to memorize. The other solutions are obtained by multiplying each term by various powers of \omega, where \omega^3=1 and \omega^2+\omega+1=0:

x=\displaystyle{\omega\left(-b+\sqrt{b^2+a^3}\right)^{\!1/3}+\omega^2\left(-b-\sqrt{b^2+a^3}\right)^{\!1/3}}

and

x=\displaystyle{\omega^2\left(-b+\sqrt{b^2+a^3}\right)^{\!1/3}+\omega\left(-b-\sqrt{b^2+a^3}\right)^{\!1/3}}.

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One response to “The quadratic formula

  1. Good info. Lucky me I came across your website by chance (stumbleupon).
    I’ve saved it for later!

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