Question

If x2 + 1/x2 = 51, find x3-1/x3
​

Answer 1

GivEn:

• x² + 1/x² = 51

To find:

• x³ - 1/x³

Solution:

⟶ x² + 1/x² = 51

⟶ (x - 1/x)² = x² + 1/x² - 2

⟶ (x - 1/x)² = 51 - 2

⟶ (x - 1/x)² = 49

⟶ (x - 1/x) = √49 = 7

___________________

Now, As we know that,

• a³ - b³ = (a - b)(a² + ab + b²)

Here,

• a = x
• b = 1/x

Substituting the values.

⟶ x³ - 1/x³ = (x - 1/x)(x² + x(1/x) + 1/x²)

By cancelling x,

⟶ x³ - 1/x³ = 7(x² + (1) + 1/x²)

⟶ x³ - 1/x³ = 7(x² + 1/x² + 1)

⟶ x³ - 1/x³ = 7(51 + 1)

⟶ x³ - 1/x³ = 7(52)

⟶ x³ - 1/x³ = 364

∴ Hence, The value of x³ - 1/x³ is 364.