If x + 1/x = 2√10, where x 1, then find the value of (x³ - 1/x³).

Question

If x + 1/x = 2√10, where x > 1, then find the value of (x³ - 1/x³).

Accepted Answer

Given: x + 1/x = 2√10, and x > 1.

We need to find x³ - 1/x³.

Step 1: Square both sides of the given equation:
(x + 1/x)² = (2√10)²
=> x² + 2 + 1/x² = 4 × 10
=> x² + 1/x² + 2 = 40
=> x² + 1/x² = 40 - 2 = 38

Step 2: Use the identity for cubes:
(x - 1/x)³ = x³ - 3x + 3(1/x) - 1/x³
But this is complicated; instead, use the identity:
(x - 1/x)³ = x³ - 3x(1/x)(x - 1/x) - 1/x³

Alternatively, use the identity:
(x³ - 1/x³) = (x - 1/x)(x² + 1 + 1/x²)

We know x + 1/x = 2√10.

Step 3: Find x - 1/x:
(x + 1/x)² - (x - 1/x)² = 4x(1/x) = 4
=> (2√10)² - (x - 1/x)² = 4
=> 40 - (x - 1/x)² = 4
=> (x - 1/x)² = 40 - 4 = 36
=> x - 1/x = ±6
Since x > 1, x - 1/x > 0, so x - 1/x = 6

Step 4: Calculate x³ - 1/x³:
Using the identity:
(x - 1/x)³ = x³ - 3x(1/x)(x - 1/x) - 1/x³
=> (x - 1/x)³ = x³ - 3(x - 1/x) - 1/x³
=> (x - 1/x)³ = (x³ - 1/x³) - 3(x - 1/x)

Rearranged:
(x³ - 1/x³) = (x - 1/x)³ + 3(x - 1/x)

Substitute x - 1/x = 6:
(x³ - 1/x³) = 6³ + 3 × 6 = 216 + 18 = 234

Therefore, the value of (x³ - 1/x³) is 234.

Answer

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