Mathematician Finds Easier Way to Solve Quadratic Equations

[Thomas Klemm]

The following geometric approach uses the right triangle altitude theorem.



Given \(c = p + q\) and \(h^2 = p \cdot q\) we get:

\(
\begin{align}
0
&=(x - p)(x - q) \\
&= x^2 - (p+q)x + p \cdot q \\
&= x^2 - c x + h^2 \\
\end{align}
\)

But then also:

\(
\begin{align}
c &= 2r \\
r^2 &= h^2 + d^2
\end{align}
\)

And therefore:

\(
\begin{align}
r &= \frac{c}{2} \\
d &= \sqrt{r^2 - h^2} \\
\\
p &= r + d \\
q &= r - d \\
\end{align}
\)

Example

\(
\begin{align}
x^2 - 5x + 6 = 0 \\
\end{align}
\)

\(
\begin{align}
c &= 5 \\
h^2 &= 6 \\
\\
r &= 2.5 \\
d
&= \sqrt{6.25 - 6} \\
&= \sqrt{0.25} \\
&= 0.5 \\
\\
p &= 2.5 + 0.5 = 3 \\
q &= 2.5 - 0.5 = 2 \\
\end{align}
\)