Derivation Of Quadratic Formula - Math Is Fun

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Derivation of Quadratic Formula

Let us find out how the famous Quadratic Formula can be created using a bunch of algebra steps.

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 withax^2 + bx + c=0 Divide the equation by ax^2 + bx/a + c/a = 0 Put c/a on other sidex^2 + bx/a = -c/a Add (b/2a)2 to both sidesx^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"

We will try to rearrange the equation to have just "x" on the left:

Start with(x+b/2a)^2 = -c/a + (b/2a)^2 Square root both sides(x+b/2a) = (+-) sqrt(-c/a+(b/2a)^2) Move b/2a to rightx = -b/2a (+-) sqrt(-c/a+(b/2a)^2) x is on its own! but let's simplify Multiply right by 2a/2ax = [ -b (+-) sqrt(-(2a)^2 c/a + (2a)^2(b/2a)^2) ] / 2a Simplify inside rootx = [ -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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