Derivation of Quadratic Formula

A Quadratic Equation looks like this:

Quadratic Equation: ax^2 + bx + c = 0

And it can be solved using the Quadratic Formula:

Quadratic Formula: x = [ -b (+-) sqrt(b^2 - 4ac) ] / 2a

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:

Start with ax^2 + bx + c=0
Divide the equation by a x^2 + bx/a + c/a = 0
Put c/a on other side x^2 + bx/a = -c/a
Add (b/2a)2 to both sides x^2 + bx/a + (b/2a)^2 = -c/a + (b/2a)^2

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" (x+b/2a)^2 = -c/a + (b/2a)^2

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

Start with (x+b/2a)^2 = -c/a + (b/2a)^2
Square root (x+b/2a) = (+-) sqrt(-c/a+(b/2a)^2)
Move b/2a to right x = -b/2a (+-) sqrt(-c/a+(b/2a)^2)

That is actually solved! But let's simplify it a bit:
Multiply right by 2a/2a x = [ -b (+-) sqrt(-(2a)^2 c/a + (2a)^2(b/2a)^2) ] / 2a
Simplify: x = [ -b (+-) sqrt(-4ac + b^2) ] / 2a


Which is the Quadratic formula we all know and love:

Quadratic Formula: x = [ -b (+-) sqrt(b^2 - 4ac) ] / 2a

 

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