STD 11 - 4.2 Quadratic equations and inequations — JEE STD 11 Science — Question

Given, $x+\frac{1}{x}=a$ and $x^2+\frac{1}{x^3}=b$

Now, squaring both sides, we get

$\qquad\left(x+\frac{1}{x}\right)^2=a^2$

$\Rightarrow \quad x^2+\frac{1}{x^2}+2=a^2$

$\text { On cubing both sides, we get }$

$\quad\left(x+\frac{1}{x}\right)^3=x^3+\frac{1}{x^3}+3\left(x+\frac{1}{x}\right)=a^3 \ldots \text { (ii) }$

$\text { On adding Eqs. (i) and (ii), we get }$

$\left(x^2+\frac{1}{x^3}\right)+\left(x^3+\frac{1}{x^2}\right)+2+3\left(x+\frac{1}{x}\right)$

$\Rightarrow \quad b+\left(x^3+\frac{1}{x^2}\right)+2+3 a=a^3+a^2$

$x^3+\frac{1}{x^2}=a^3+a^2-3 a-b-2$