For instance, for the class of quadratic polynomial equations
( 
)
recall that even if you cannot factor the equation, it is still
possible to apply the Quadratic Formula:
to find the roots. Depending on your class, you may or may have not seen why this works. One standard way to derive the quadratic formula uses several of the manipulation techniques described above to rewrite the quadratic polynomial in the following equivalent form. First factoring out the leading coefficient:
then completing the square:
we get another form of the original equation. Solving the original equation is the same as solving
We can do this by taking square roots
Then additional simplifications yield
so
