Question

If x - 1 / x = 5 , find the value of x^2 + 1/ x^2 and x ^4 + 1/x^4

Answer 1

$\sf x-\dfrac{1}{x}=5$

On squaring both sides,

$\implies \sf \left( x-\dfrac{1}{x} \right)^2=5^2$

$\bullet\ \; \qquad \bf (a-b)^2=a^2-2ab+b^2$

$\implies \sf x^2- 2.x.\dfrac{1}{x}+ \left( \dfrac{1}{x} \right)^2=25$

$\implies \sf x^2-2+\dfrac{1}{x^2}=25$

$\implies \sf x^2+\dfrac{1}{x^2} =25+2$

$\implies \sf \pink{x^2+\dfrac{1}{x^2} =27}$

Again squaring both sides,

$\implies \sf \left(x^2+\dfrac{1}{x^2} \right)^2=(27)^2$

$\bullet\ \; \qquad \bf (a+b)^2=a^2+2ab+b^2$

$\\ \implies \sf (x^2)^2+ 2.x^2.\dfrac{1}{x^2}+ \left(\dfrac{1}{x^2} \right)^2 =(27)^2$

$\implies \sf x^4+2+\dfrac{1}{x^4}=729$

$\implies \sf x^4+\dfrac{1}{x^4}=729-2$

$\implies \sf \blue{x^4+\dfrac{1}{x^4}=727}$