Question

If a3 3a2 9a=1 then find the value of a3 3/a

Answer 1

Given

$a^3+3a^2+9a=1$

To find

$a^3+\dfrac{3}{a}$

Solution

$a^3+3a^2+9a=1\\\Rightarrow a^3=1-3a^2-9a\quad ...(i)$

$a^3+3a^2+9a=1\\\Rightarrow a(a^2+3a+9)=1\\\Rightarrow a^2+3a+9=\dfrac{1}{a}\\\Rightarrow \dfrac{3}{a}=3(a^2+3a+9)\quad ...(ii)$

From $(i)$ and $(ii)$

$a^3+\dfrac{3}{a}=3a^2-9a+3(a^2+3a+9)\\\Rightarrow a^3+\dfrac{3}{a}=1-3a^2-9a+3a^2+9a+27\\\Rightarrow a^3+\dfrac{3}{a}=27+1\\\Rightarrow a^3+\dfrac{3}{a}=28$

$\boldsymbol{\therefore a^3+\dfrac{3}{a}=28}$