We want to rewrite the general quadratic equation $ax^2 + bx + c$ in the form $(\alpha x + \beta)(\gamma x + \delta)$.

To do this we look for two numbers $p$ and $q$ such that $ac = pq$ and $p + q = b$.

We can now rewrite the general quadratic equation as …

Now, since $ac = pq$ then $\frac{a}{p} = \frac{q}{c}$ and thus also $\frac{a}{p}x + 1 = \frac{q}{c}x + 1$.

Therefore we can further write the general quadratic equation as …

Let’s try some examples.

Example 1

$a = 6$, $b = 11$ and $c = 3$; $ac = 6 \cdot 3 = 18 = 2 \cdot 9$ and $b = 11 = 2 + 9$; so $p = 2$. Therefore …

Example 2

$a = 6$, $b = 13$ and $c = 6$; $ac = 6 \cdot 6 = 36 = 4 \cdot 9$ and $b = 13 = 4 + 9$; so $p = 4$. Therefore …

Example 3

$a = 9$, $b = 12$ and $c = -12$; $ac = 9 \cdot -12 = -108 = -6 \cdot 18$ and $b = 12 = -6 + 18$, so $p = -6$. Therefore …