Solve the equation x^4 - 2x^2 +1 = 0 for x, where x is real. What on earth do i do? How do i start!
Thanks in advance
Let's look at your original equation:
note that
This means that your original equation can be rewritten as
What Prove It did was to substitute for .
If you substitute for in the equation above,
you will get this:
This equation can be solved for with the quadratic formula, as it's a quadratic equation with as an unknown, and as we know that stands for , this means that If we know we can get that , and then solve for when we have figured out .
This is a longer method but the more ways you know how to do something the better! Alternatively, the factor theorem could be used.
Subbing x = 1 into the quartic gives you 0 so x-1 is a factor. Using long division we can get a cubic answer. Using the factor theorem again on that will get you a factorisable quadratic.
Then you will have 4 factors!