Question

if x - 1/x =3+2√2 , find the value of 1/4(x³-1/x³)​

Answer 1

EXPLANATION.

⇒ (x - 1/x) = 3 + 2√2.

Cubing on both sides of the equation, we get.

⇒ (x - 1/x)³ = (3 + 2√2)³.

As we know that,

Formula of :

⇒ (x - y)³ = x³ - 3x²y + 3xy² - y³.

⇒ (x + y)³ = x³ + 3x²y + 3xy² + y³.

Using this formula in the equation, we get.

⇒ (x³) - 3(x²)(1/x) + 3(x)(1/x)² - (1/x)³ = [(3)³ + 3(3)²(2√2) + 3(3)(2√2)² + (2√2)³].

⇒ (x³) - 3x + 3/x - 1/x³ = [27 + 27(2√2) + 9(8) + 16√2)].

⇒ (x³) - 3(x - 1/x) - 1/x³ = [27 + 54√2 + 72 + 16√2].

Put the value of x - 1/x = 3 + 2√2 in the equation, we get.

⇒ x³ - 3(3 + 2√2) - 1/x³ = [99 + 70√2].

⇒ x³ - 9 - 6√2 - 1/x³ = [99 + 70√2].

⇒ x³ - 1/x³ = [99 + 70√2] + 9 + 6√2.

⇒ x³ - 1/x³ = 99 + 6 + 70√2 + 6√2.

⇒ x³ - 1/x³ = 105 + 76√2.

To find :

⇒ 1/4 (x³ - 1/x³).

⇒ 1/4 x (105 + 76√2).

⇒ 1/4 (x³ - 1/x³) = (105 + 76√2)/4.