Question

if x(x-2)=1,then the value of x^2+1/x^2​

Answer 1

Given,

$\longrightarrow\sf{x(x-2)=1}$

Expanding LHS,

$\longrightarrow\sf{x^2-2x=1}$

Adding $\sf{2x-1}$ to both sides,

$\longrightarrow\sf{x^2-1=2x}$

Dividing by $\sf{x,}$

$\longrightarrow\sf{\dfrac{x^2-1}{x}=2}$

$\longrightarrow\sf{x-\dfrac{1}{x}=2}$

Squaring both sides,

$\longrightarrow\sf{\left(x-\dfrac{1}{x}\right)^2=2^2}$

$\longrightarrow\sf{x^2-2\cdot x\cdot\dfrac{1}{x}+\dfrac{1}{x^2}=4}$

$\longrightarrow\sf{x^2-2+\dfrac{1}{x^2}=4}$

Adding 2 to both sides, we get,

$\longrightarrow\underline{\underline{\sf{x^2+\dfrac{1}{x^2}=6}}}$

Hence 6 is the answer.