Question

Find x^(2)+(1)/(x^(2)) and x^(4)+(1)/(x^(4) if x-1/x =9 ​

Answer 1

Answer :

• The value of x² + 1/x² = 83
• The value of x⁴ + 1/x⁴ = 6887

Explanation :

Given :

• The value of x - 1/x = 9 ⠀⠀⠀⠀Eq.(i)

To find :

• The value of x² + 1/x² = ?
• The value of x⁴ + 1/x⁴ = ?

Knowledge required :

• (a - b)² = a² + b² - 2ab
• (a + b)² = a² + b² - 2ab

Solution :

By squaring on both the sides of the equation.(i), we get :

$:\implies \sf{x - \dfrac{1}{x} = 9} \\ \\ :\implies \sf{\bigg(x - \dfrac{1}{x}\bigg)^{2} = 9^{2}} \\ \\ :\implies \sf{x^{2} + \dfrac{1}{x^{2}} - 2 \times x \times \dfrac{1}{x} = 9^{2}} \\ \\ :\implies \sf{x^{2} + \dfrac{1}{x^{2}} - 2 \times \not{x} \times \dfrac{1}{\not{x}} = 81} \\ \\ :\implies \sf{x^{2} + \dfrac{1}{x^{2}} - 2 = 81} \\ \\ :\implies \sf{x^{2} + \dfrac{1}{x^{2}} = 81 + 2} \\ \\ :\implies \sf{x^{2} + \dfrac{1}{x^{2}} = 83} \\ \\ \boxed{\therefore \sf{x^{2} + \dfrac{1}{x^{2}} = 83}}$

From the above equation, we get :

• The value of x² + 1/x² = 83 ⠀⠀⠀⠀Eq.(ii)

By squaring on both the sides of the equation ii , we get :

$:\implies \sf{\bigg(x^{2} - \dfrac{1}{x^{2}}\bigg)^{2} = 83^{2}} \\ \\ :\implies \sf{(x^{2})^{2} - \dfrac{1}{(x^{2})^{2}} + 2 \times x^{2} \times \dfrac{1}{x^{2}} = 6889} \\ \\ :\implies \sf{(x^{2})^{2} - \dfrac{1}{(x^{2})^{2}} + 2 \times \not{x^{2}} \times \dfrac{1}{\not{x^{2}}} = 6889} \\ \\ :\implies \sf{(x^{2})^{2} - \dfrac{1}{(x^{2})^{2}} + 2 = 6889} \\ \\ :\implies \sf{x^{4} - \dfrac{1}{x^{4}} = 6889 - 2} \\ \\ :\implies \sf{x^{4} - \dfrac{1}{x^{4}} = 6887} \\ \\ \boxed{\therefore \sf{x^{4} + \dfrac{1}{x^{4}} = 6887}}$

Therefore,

• The value of x² + 1/x² = 83
• The value of x⁴ + 1/x⁴ = 6887