Why can ALL quadratic equations be solved by the quadratic formula? In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use it. I have tried to figure it out by proving these two equations are equal, but I can't.

Why can I use $x = \dfrac{-b\pm \sqrt{b^{2} - 4 ac}}{2a}$ to solve all quadratic equations?
I would like to prove the Quadratic Formula in a cleaner way.  Perhaps if teachers see this approach they will be less reluctant to prove the Quadratic Formula.

Added: I have recently learned from the book Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century (Ranjan Roy) that the method described below was used by the ninth century mathematician Sridhara. (I highly recommend Roy's book, which is much broader in its coverage than the title would suggest.)

We want to solve the equation
$$ax^2+bx+c=0,$$
where $a \ne 0$.  The usual argument starts by dividing by $a$.  That is a strategic error, division is ugly, and produces formulas that are unpleasant to typeset.  

Instead, multiply both sides by $4a$.  We obtain the equivalent equation
$$4a^2x^2 +4abx+4ac=0.\tag{1}$$
Note that $4a^2x^2+4abx$ is almost the square of $2ax+b$. More precisely,
$$4a^2x^2+4abx=(2ax+b)^2-b^2.$$
So our equation can be rewritten as
$$(2ax+b)^2 -b^2+4ac=0 \tag{2}$$
or equivalently
$$(2ax+b)^2=b^2-4ac. \tag{3}$$
Now it's all over. We find that
$$2ax+b=\pm\sqrt{b^2-4ac} \tag{4}$$
and therefore
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}. \tag{5}$$
No fractions until the very end!

Added: I have tried to show that initial division by $a$, when followed by a completing the square procedure, is not a simplest  strategy. One might remark additionally that if we first divide by $a$, we end up needing a couple of additional "algebra" steps to partly undo the division in order to give the solutions their traditional form.

Division by $a$ is definitely a right beginning if it is followed by an argument that develops the connection between the coefficients and the sum and product of the roots.  Ideally, each type of proof should be presented, since each connects to an important family of ideas.  And a twice proved theorem is twice as true.
If one is interested only in the roots one can concentrate on the monic equations. Now any monic quadratic expression becomes the square of a linear expression by addition of a suitable constant (finding what the suitable constant  is called  completing the square).  The additive constant is uniformly the same expression in the coefficients of the given quadratic equation. 

So the given equation has the same solutions as the modified equation of the form $(x+B)^2=A$. SO the formula for the roots are of the form $x=-B \pm \sqrt A$. Thats the reason all quadratic equations have solutions expressed in a uniform manner.