Question

X2 + 1/x2 = 7 find x +1/x and x - 1/x​

Answer 1

       $\underline{\bold{Answer:}}$

       $\boxed{x + \frac{1}{x} = +3,-3 }, \text{ }\boxed{x-\frac{1}{x} = \sqrt5}$

       $\underline{\bold{Step-by-step-explaination:}}$

       $\text{The given equation is: }$

       $\boxed{x^{2} + \frac{1}{x^{2}} = 7} \text{ ------(i)}$

       $\text{We know that }(a+b)^{2} = a^{2} + b^{2} + 2ab$

$\implies \boxed{(x + \frac{1}{x})^{2} = (x)^{2} + (\frac{1}{x})^{2} + 2(x)(\frac{1}{x}) }$

$\implies \boxed{(x + \frac{1}{x})^{2} = x^{2} + \frac{1}{x^{2}} + 2 }$

       $\text{From (i)}$

$\implies \boxed{(x + \frac{1}{x})^{2} = 7 + 2}$

$\implies \boxed{(x + \frac{1}{x})^{2} = 9}$

$\implies \boxed{x + \frac{1}{x} = \sqrt{9} }$

$\implies \boxed{x + \frac{1}{x} = +3,-3 }$

       $\text{We know that }(a-b)^{2} = a^{2} + b^{2} - 2ab$

$\implies \boxed{(x-\frac{1}{x})^{2} = (x)^{2} + (\frac{1}{x})^{2}- 2(x)(\frac{1}{x} ) }$

$\implies \boxed{(x-\frac{1}{x})^{2} = x^{2} + \frac{1}{x^{2}} - 2}$

       $\text{From (i)}$

$\implies \boxed{(x-\frac{1}{x})^{2} = 7-2}$

$\implies \boxed{(x-\frac{1}{x})^{2} = 5}$

$\implies \boxed{x-\frac{1}{x} = \sqrt5}$