Question

If x^2+1/x^2=3 find the value of x^2/(x^2+1)^2

Express answer as a common fraction.
Thanks!!

Answer 1

`x^2+\frac1{x^2}=3\implies x^4+1=3x^2\implies x^4-3x^2+1=0`

By the quadratic formula,

`x^2=\frac{3\pm\sqrt5}2\implies x^2+1=\frac{5\pm\sqrt5}2`

Then

`(x^2+1)^2=\frac{25\pm10\sqrt5+5}4=\frac{15\pm5\sqrt5}2`

`\implies(x^2)/((x^2+1)^2)=\frac{\frac{3\pm\sqrt5}2}{\frac{15\pm5\sqrt5}2}=(3\pm\sqrt5)/(15\pm5\sqrt5)`

Multiply numerator and denominator by the denominator's conjugate:

`(3\pm\sqrt5)/(15\pm5\sqrt5)\cdot(15\mp5\sqrt5)/(15\mp5\sqrt5)=(45\pm15\sqrt5\mp15\sqrt5-25)/(15^2-(5\sqrt5)^2)=(20)/(100)=\frac15`