Question

(x+1/x) = 6 find the value of (x-1/x)​

Answer 1

ANSWER✔

$\large\underline\bold{GIVEN,}$

$\sf\dashrightarrow x+ \dfrac{1}{x } =6$

$\large\underline\bold{TO\:FIND,}$

$\sf\dashrightarrow THE\:VALUE\:OF \: x- \dfrac{1}{x}$

✯IDENTITY IN USE,

$\large{\boxed{\bf{ \star\:\:(a+b)^2=a^2+b^2+2ab \:\: \star}}}$

$\large{\boxed{\bf{ \star\:\: (a-b)^2=a^2+b^2-2ab\:\: \star}}}$

$\large\underline\bold{SOLUTION,}$

$\sf\therefore x+ \dfrac{1}{x } =6$

$\purple{\text{ squaring both sides}}$

$\sf\implies \bigg( x+ \dfrac{1}{x}\bigg)^2 =(6 )^2$

$\sf\implies \bigg( x^2+ \dfrac{1}{x^2} +2\bigg)= 36$

$\sf\implies x^2+ \dfrac{1}{x^2} =36-2$

$\sf\implies x^2+ \dfrac{1}{x^2} =34 \: -----\boxed{equation\:1}$

NOW,

TAKING,

$\sf\therefore x- \dfrac{1}{x}$

$\purple{\text{ squaring the equation}}$

$\sf\implies \bigg( x- \dfrac{1}{x}\bigg)^2= x^2 + \dfrac{1}{x^2}-2$

$\sf\implies \bigg( x- \dfrac{1}{x}\bigg)^2= 34-2 \: -----\boxed{from\: equation\:1}$

$\sf\implies \bigg( x- \dfrac{1}{x}\bigg)^2= 34-2$

$\sf\implies \bigg( x- \dfrac{1}{x}\bigg)^2= 32$

$\sf\implies x- \dfrac{1}{x}= \pm \sqrt{32}$

$\large{\boxed{\bf{ \star\:\: x- \dfrac{1}{x}= \pm \sqrt{32}\:\: \star}}}$

$\large\underline\bold{THE\:VALUE\:OF \: x- \dfrac{1}{x}\:IS\: \pm \sqrt{32}}$

______________________

Answer 2

$\huge\underline{\red{⛥❃꧁αղsw៩Ʀ꧂⛥❃}}$

$\large\underline\bold{GIVEN,}$

$\sf\dashrightarrow x+ \dfrac{1}{x } =6$

$\large\underline\bold{TO\:FIND,}$

$\sf\dashrightarrow THE\:VALUE\:OF \: x- \dfrac{1}{x}$

$\large{\boxed{\bf{ \star\:\:(a+b)^2=a^2+b^2+2ab \:\: \star}}}$

$\large{\boxed{\bf{ \star\:\: (a-b)^2=a^2+b^2-2ab\:\: \star}}}$

$\large\underline\bold{SOLUTION,}$

$\sf\therefore x+ \dfrac{1}{x } =6$

$\purple{\text{ squaring both sides}}$

$\sf\implies \bigg( x+ \dfrac{1}{x}\bigg)^2 =(6 )^2$

$\sf\implies \bigg( x^2+ \dfrac{1}{x^2} +2\bigg)= 36$

$\sf\implies x^2+ \dfrac{1}{x^2} =36-2$

$\sf\implies x^2+ \dfrac{1}{x^2} =34 \: -----\boxed{equation\:1}$

NOW,

TAKING,

$\sf\therefore x- \dfrac{1}{x}$

$\purple{\text{ squaring the equation}}$

$\sf\implies \bigg( x- \dfrac{1}{x}\bigg)^2= x^2 + \dfrac{1}{x^2}-2$

$\sf\implies \bigg( x- \dfrac{1}{x}\bigg)^2= 34-2 \: -----\boxed{from\: equation\:1}$

$\sf\implies \bigg( x- \dfrac{1}{x}\bigg)^2= 34-2$

$\sf\implies \bigg( x- \dfrac{1}{x}\bigg)^2= 32$

$\sf\implies x- \dfrac{1}{x}= \pm \sqrt{32}$

$\large{\boxed{\bf{ \star\:\: x- \dfrac{1}{x}= \pm \sqrt{32}\:\: \star}}}$

$\large\underline\bold{THE\:VALUE\:OF \: x- \dfrac{1}{x}\:IS\: \pm \sqrt{32}}$

______________________