A simpler form of the quadratic formula

In high school, I had to learn the famous quadratic formula by heart. It states that a quadratic polynomial pp defined as
p(x)=ax2+bx+cp(x) = ax^2 + bx +c
has roots x1,2x_{1, 2} of the form
x1,2=−b±b2−4ac2ax_{1, 2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

I think remembering formulae is a prime example of how math should not be taught. I think the proper way to solve quadratic equations is to complete the square:

First, rewrite the equation to the form
x2+bx+c=0x^2 + bx + c = 0

We can rewrite this as
(x−b2)2−(b2)2+c=0 (x - \frac{b}{2})^2 - (\frac{b}{2})^2 + c = 0

From there, it is easy to take it to
(x−b2)2=(b2)2−c(x - \frac{b}{2})^2 = (\frac{b}{2})^2 - c

and solve the resulting equation, remembering that x2=cx^2 = c has two solutions: x=cx = \sqrt{c} and x=−cx = -\sqrt{c}:

x−b2=±(b2)2−cx - \frac{b}{2} = \pm \sqrt{(\frac{b}{2})^2 - c}

So x1,2=b2±(b2)2−cx_{1, 2} = \frac{b}{2} \pm \sqrt{(\frac{b}{2})^2 - c}.

If you, for some reason, insist on learning a formula (for example, because it is easier or faster to apply), there is an alternative that I find to be preferable to the classical quadratic formula. By setting a=−q2pa = -\frac{q}{2p} and b=−rpb = -\frac{r}{p}, any quadratic equation of the form px2+qx+r=0px^2 + qx + r = 0 can be reduced to the form
x2=2ax+b x^2 = 2ax + b

which has the solution
x1,2=a±a2+b x_{1, 2} = a \pm \sqrt{a^2 + b}

and is far more elegant, in my opinion.

In conclusion, I think it’s preferable to not teach the quadratic formula in high school, and instead teach how to complete the square. Not only does this relieve you of the burden of remembering a formula, it is also much more instructive and opens doors to more elaborate algebraic manipulation.