SSC IMD SA 2017 – Question 37

If a³ + b³ = 208 and ab = -12, then what is the value of a + b?

• 8
• 12
• 11
• 4

Detailed Explanation:

This algebraic problem utilizes the identity for the sum of two cubes.

Step-by-step Solution:

1. Given Data:
   • $a^3 + b^3 = 208$
   • $ab = -12$
2. Algebraic Identity:$$a^3 + b^3 = (a + b)^3 – 3ab(a + b)$$
3. Let $(a + b) = s$. Substitute the known values into the identity:$$208 = s^3 – 3(-12)s$$$$208 = s^3 + 36s$$$$s^3 + 36s – 208 = 0$$
4. Testing the Options:
   • For $s = 4$:$$4^3 + 36(4) = 64 + 144 = 208$$This satisfies the equation perfectly!
   • For $s = 8$: $8^3 + 36(8) = 512 + 288 = 800 \neq 208$.
5. Thus, $s = a + b = 4$.

Therefore, the value of $a + b$ is 4. The correct option is d).