# Quadratic Equations

In Algebra we learned the solution to the quadratic equation

is

But, if you are like me, we did not learn how to derive that solution. Read on to see one approach.

## Simplify

To start, simplify the quadratic equation by dividing both sides by 'a'. This does not change the values of x that solve this equation. The equeation becomes

Replace b/a with B and c/a with C, so now we are looking for the solution to the following.

## Factoring

This simplified quadratic equation

can be factored to

where and are the two solution values for x. Multiplying this out, we have

So, to derive the solution to the simplified quadratic equation, we need to solve the following simultaneous equations for and as a function of B and C.

## First Try

This looks easy. Lets solve for using the first equation, and substitute the solution for into the second equation.

simplifying, we have

Oh no, we are back to solving the original problem. This approach is a dead end.

## Second Try

Remember, we are trying to solve the following equations for and :

Let's create a new equation by squaring both sides of the first equation and then substituting C for .

And, now let's solve these 2 equations using geometry.

Referring to the following diagram, the blue triangle represents the first equation, and the green line represents the second.

Regarding this daigram, we know what P and Q are. This leads to R, which leads to S, which leads finally to .

This can be simplified to

can be determined from

We now have the two values of x that solve the simplified quadratic equation.

Replacing B with b/a, and C with c/a we have the solution to the quadratic equation