In Algebra we learned the solution to the quadratic equation
But, if you are like me, we did not learn how to derive that solution. Read on to see one approach.
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.
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.
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.
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