Question

If x-1/X is 4, then find x²+1/x² and x⁴+1/x⁴

Answer 1

$\bf{\underline{Given}}$

$\sf{x-\dfrac{1}{x} = 4}$

$\bf{\underline{To\:Find}}$

• $\sf{ x^2 + \dfrac{1}{x^2}}$

• $\sf{x^4 + \dfrac{1}{x^4}}$

$\bf{\underline{Solution}}$

$\sf{x-\dfrac{1}{x} = 4}$

$\sf{\underline{By\:squaring\:both\:the\:sides}}$

= $\sf{\bigg(x-\dfrac{1}{x}\bigg)^2 = (4)^2}$

= $\sf{x^2 - 2\times x\times \dfrac{1}{x} + \dfrac{1}{x^2} = 16}$

= $\sf{x^2 - 2 + \dfrac{1}{x^2} = 16}$

= $\sf{x^2 + \dfrac{1}{x^2} = 16+2}$

= $\sf{x^2 + \dfrac{1}{x^2} = 18}$

$\sf{\therefore x^2 + \dfrac{1}{x^2} = 18}$

Now,

We have,

$\sf{x^2 + \dfrac{1}{x^2} = 18}$

$\sf{\underline{By\:squaring\:both\:the\:sides}}$

= $\sf{\bigg(x^2 + \dfrac{1}{x^2}\bigg)^2 = (18)^2}$

= $\sf{(x^2)^2 + 2\times x^2 \times\dfrac{1}{x^2} + \bigg(\dfrac{1}{x^2}\bigg)^2 = 324}$

= $\sf{x^4 + 2 + \dfrac{1}{x^4} = 324}$

= $\sf{x^4 + \dfrac{1}{x^4} = 324 - 2}$

= $\sf{x^4 + \dfrac{1}{x^4} = 322}$

$\sf{\therefore x^4 + \dfrac{1}{x^4} = 322}$