Question

x=1/5-x find x³+1/x³

Answer 1

Answer :-

x³ + 1/x³ = 110

Explanation :-

Given :-

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

To find :-

x³ + 1/x³

Solution :-

First find x + 1/x

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

Reciprocal on both sides

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

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

Transpose - x to RHS [ - x becomes + x]

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

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

$\sf \therefore x + \dfrac{1}{x} = 5$

Cubing on both sides

(x + 1/x)³ = (5)³

⇒ (x + 1/x)³ = 125

We know that

(a + b)³ = a³ + b³ + 3ab(a + b)

Here a = x, b = 1/x

By substituting the values

⇒ (x)³ + (1/x)³ + 3(x)(1/x)(x + 1/x) = 125

⇒ x³ + 1³/x³ + 3(x + 1/x) = 125

⇒ x³ + 1/x³ + 3(x + 1/x) = 125

⇒ x³ + 1/x³ + 3(5) = 125

⇒ x³ + 1/x³ + 15 = 125

⇒ x³ + 1/x³ = 125 - 15

⇒ x³ + 1/x³ = 110

Therefore the value of x³ + 1/x³ is 110.

Identity used :-

• (a + b)³ = a³ + b³ + 3ab(a + b)