Question

If x^3 + 1/x^3 = 110 then find x + 1/x

Answer 1

`Use:(a+b)^3=a^3+3a^2b+3ab^2+b^3\ (*)\\\\x^3+\left((1)/(x)\right)^3\\=\underbrace{x^3+3\cdot x^2\cdot(1)/(x)+3\cdot x\cdot\left((1)/(x)\right)^2+\left((1)/(x)\right)^3}_((*))-3\cdot x^2\cdot(1)/(x)-3\cdot x\cdot\left((1)/(x)\right)^2\\\\=\left(x+(1)/(x)\right)^3-3x-3\cdot(1)/(x)=\left(x+(1)/(x)\right)^3-3\left(x+(1)/(x)\right)`

therefore


`x^3+\left((1)/(x)\right)^3=110\iff \left(x+(1)/(x)\right)^3-3\left(x+(1)/(x)\right)=110\\\\subtitute\ t=x+(1)/(x)\\\\t^3-3t=110\ \ \ \ |subtract\ 110\ from\ both\ sides\\\\t^3-3t-110=0\\\\t^3-25t+22t-110=0\\\\t(t^2-25)+22(t-5)=0\ \ \ |use\ a^2-b^2=(a-b)(a+b)\\\\t(t-5)(t+5)+22(t-5)=0\\\\(t-5)[t(t+5)+22]=0\iff t-5=0\ or\ \underbrace{t^2+5t+22=0}_(no\ solution)\\\\t=5\iff x+(1)/(x)=5\\\\Answer:\boxed{x+(1)/(x)=5}`