[SOLVED] Factoring a quadratic

Can someone please tell me why this works? (It works in general apparently.)

Let's factor

$\displaystyle 12x^2 - 25x + 12$

Take the leading coefficient (12) and multiply it into the constant and rewrite this as

$\displaystyle x^2 - 25x + 144$

Now factor it:

$\displaystyle (x - 9)(x - 16)$

Now divide both of the constants by that same 12 as before:

$\displaystyle \left ( x - \frac{9}{12} \right ) \left ( x - \frac{16}{12} \right )$

$\displaystyle = \left ( x - \frac{3}{4} \right ) \left ( x - \frac{4}{3} \right )$

Now multiply the whole thing by that same 12:

$\displaystyle 12 \cdot \left ( x - \frac{3}{4} \right ) \left ( x - \frac{4}{3} \right )$

$\displaystyle = 4 \cdot \left ( x - \frac{3}{4} \right ) \cdot 3 \cdot \left ( x - \frac{4}{3} \right )$

$\displaystyle = (4x - 3)(3x - 4)$

which is the factored form of the original problem.

I can't figure out why this works???!!

-Dan