(2x) / (1+x^2)^2 = 2/5
I completely forgot how to solve equations with x^4
can someone tell me how to do this?
I take the denominator to the other side and factored it and got an equation with x to the 4th degree.
what was the equation you got? show us and we'll take it from there, there is no one way to attack a quartic, we must do it by case
EDIT: ok, this is not a "nice" quadratic, it does not have integer roots. so if you have to solve this by hand i see no other way to do it at the moment other than to use the quartic formula. which is a mess. you can use Cardano's or Galois' method (your professor is cruel)
Let $\displaystyle x=\tan y$ then we have,
$\displaystyle \frac{2\tan y}{(1+\tan^2 y)^2} = \frac{2}{5}$
Thus,
$\displaystyle \frac{2\tan y}{\sec^4 y} = \frac{2}{5}$
Thus,
$\displaystyle 2\tan y\cos ^4 y = \frac{2}{5}$
Which means,
$\displaystyle 2 \sin y \cos y \cos^2 y = \frac{2}{5}$
Identities,
$\displaystyle \sin 2y \cos^2 y = \frac{2}{5}$
More,
$\displaystyle (1 - 2\cos^2 y)(\cos^2 y) = \frac{2}{5}$
Let $\displaystyle z=\cos^2 y$ to get,
$\displaystyle (1-2z)z = \frac{2}{5}$.
The above is quadradic which is solvable.
Then you need to convert $\displaystyle \cos $ into $\displaystyle \tan $.
And then take arctangent to get final answer.