Quadratic Equations – Galois Theory



In a Previous Post, we saw how to solve the Quadratic Equation using elementary methods by shifting around parabolas. Here we’ll see a more modern method that will not only help with higher degrees, but also show that equations of degree larger than four cannot be solved.

The general equation that we want to solve is $latex x^2+Bx+C=0$. If we let the coefficients be fractions, then these include the equations $latex Ax^2+Bx+C=0$, since I can just divide through by $latex A$.

1.) Playing with Roots

Let’s pretend that I have found the solutions to $latex x^2+Bx+C=0$ to be $latex x=R$ and $latex x=S$. This means that I can factor to rewrite the equation as

$latex (x-R)(x-S)=0$

If I expand the factored equation and set it equal to the original I get

$latex x^2 +Bx+C =x^2-(R+S)x+RS$

Which means that we can relate the roots and coefficients by

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