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:
- ax2 + bx + c = 0
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"
ax2 + bx + c = 0
- ∴ ax2 + bx = -c
- ∴ a2x2 + abx = -ac
- ∴ a2x2 + 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.