Question

If a – b = 2 and ab = 3, find the value of a³ + b³.

Answer 1

Given: a – b = 2 and ab = 3

Step-wise calculation:

1. We want to find (a³ + b³).

2. Use the identity for the difference of cubes:

a³ – b³ = (a – b)(a² + ab + b²)

3. Calculate (a³ + b³) by expressing it in terms of (a + b) and (ab): 

a³ + b³ = (a + b)³ – 3ab(a + b)

4. First, find (a + b) using: 

(a – b)² = a² – 2ab + b² 

⇒ (a – b)² = (a + b)² – 4ab

Given (a – b = 2),

So, 2² = (a + b)² – 4 × 3

4 = (a + b)² – 12 

(a + b)² = 16 

a + b = ±4

5. Substitute into the formula for (a³ + b³):

a³ + b³ = (a + b)³ – 3ab(a + b)

a³ + b³ = ±4³ – 3 × 3 × (±4)

If (a + b = 4):

a³ + b³ = 64 – 36

a³ + b³ = 28

If (a + b = –4): 

a³ + b³ = –64 + 36

a³ + b³ = –28