Express p in terms of q if p = {x}^(2) + \frac{1}{ {x}^(2) } \: and \: \: q = x + (1)/(x)

Question

Express p in terms of q if


`p = {x}^(2) + \frac{1}{ {x}^(2) } \: and \: \: q = x + (1)/(x)`

Answer 1

`p=x^2+(1)/(x^2)\\\\p=(x^4)/(x^2)+(1)/(x^2)\\\\p=(x^4+1)/(x^2)\qquad(*)`

`q=x+(1)/(x)\\\\q=(x^2)/(x)+(1)/(x)\\\\q=(x^2+1)/(x)\qquad\text{square both sides}\\\\q^2=\left((x^2+1)/(x)\right)^2\\\\q^2=((x^2+1)^2)/(x^2)\qquad\text{use}\ \ (a+b)^2=a^2+2ab+b^2\\\\q^2=((x^2)^2+2(x^2)(1)+1^2)/(x^2)\\\\q^2=(x^4+2x^2+1)/(x^2)\\\\q^2=(x^4+1)/(x^2)+(2x^2)/(x^2)\\\\q^2=(x^4+1)/(x^2)+2\qquad\text{subtract 2 from both sides}\\\\q^2-2=(x^4+1)/(x^2)\\\\\text{From (*) we have}\\\\\boxed{p=q^2-2}`