Derivation of Quadratic Formula
A Quadratic Equation looks like this:
And it can be solved using the Quadratic Formula:
That formula looks like magic, but you can follow the steps to see how it comes about.
1. Complete the Square
ax2 + bx + c has "x" in it twice, which is hard to solve.
But there is a way to rearrange it so that "x" only appears once. It is called Completing the Square (please read that first!).
Our aim is to get something like x2 + 2dx + d2, which can then be simplified to (x+d)2
So, let's go:
|Divide the equation by a|
|Put c/a on other side|
|Add (b/2a)2 to both sides|
The left hand side is now in the x2 + 2dx + d2 format, where "d" is "b/2a"
So we can re-write it this way:
|"Complete the Square"|
Now x only appears once and we are making progress.
2. Now Solve For "x"
Now we just need to rearrange the equation to leave "x" on the left
|Move b/2a to right|
That is actually solved! But let's simplify it a bit:
|Multiply right by 2a/2a|
Which is the Quadratic formula we all know and love: