30122009 Solving Quadratic Equations
When we're in high school, one of the equations we're told to memorize is the one for solving quadratic formulas. The rule we're taught is:
if:
 ax^{2} + bx + c = 0
then:

x = b ± √ b2  4ac2a
And this is indeed true. Sadly, what many of us are not taught (and I count myself in that unfortunate group), is why that is the case. As described in my earlier blog about the volume of a sphere, it was always a frustration of mine simply being told to memorize these things without being given a real understanding of how they work. To remedy that, I gave myself a quick mental exercise this evening, trying to reach the latter equation given the former one. It turned out much easier than I thought, and here's a breakdown of how it works.
We'll start by converting the left hand of the equation into a perfect square, a technique called "completing the square"

ax^{2} + bx + c = 0
 ∴ ax^{2} + bx = c
 ∴ a^{2}x^{2} + abx = ac
 ∴ a^{2}x^{2} + abx + (^{b}/_{2})^{2} = ac + (^{b}/_{2})^{2}
Those last two steps are the key to this solution. By multiplying both sides of the equation by "a", we turn the first term into a perfect square. By adding (^{b}/_{2})^{2} to both sides, we create a term who's square root is equal to half of "b". This makes the left side of our equation a perfect square, allowing us to reduce to a single term with the variable x:
 ∴ (ax + ^{b}/_{2})^{2} = ac + (^{b}/_{2})^{2}
 ∴ (ax + ^{b}/_{2})^{2} = ac + ^{b2}/_{4}
∴ (ax + ^{b}/_{2})^{2} = b2  4ac4∴ ax + ^{b}/_{2} = ± √ b2  4ac2∴ ax = b ± √ b2  4ac2∴ x = b ± √ b2  4ac2a
It's a really great feeling to be able to find the explanation and understand it without looking it up.