Question

If a + b + c = 9 and a³ + b³ + c³ – 3abc = 27 then, what is the value of ab + bc + ca? 
1. 24
2. 26
3. 28
4. 22

Answer 1

Correct Answer - Option 2 : 26

Given:

a + b + c = 9

a³ + b³ + c³ – 3abc = 27

Formula Used:

a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Calculations:

Here,  

a + b + c = 9

and a³ + b³ + c³ – 3abc = 27

Using the identity,

a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)

we have,

27 = 9 × (a² + b² + c² – ab – bc – ca)

⇒ 27/9 = a² + b² + c² – (ab + bc + ca)

⇒ a² + b² + c² – (ab + bc + ca) = 3

⇒ (a + b + c)² – 2(ab + bc + ca) – (ab + bc + ca) = 3      ----{∵ (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)}

⇒ (a + b + c)² – 3(ab + bc + ca) = 3

⇒ 9² – 3(ab + bc + ca) = 3

⇒ 3(ab + bc + ca) = 9² – 3

⇒ ab + bc + ca = (81 – 3)/3

⇒ ab + bc + ca = 78/3

⇒ ab + bc + ca = 26

∴ The value of ab + bc + ca is 26.