If a + b + c = 5 and ab + bc + ca = 10, then prove that a³ + b³ + c³ - 3abc = - 25. Class 9

If a + b + c = 5 and ab + bc + ca = 10, then prove that a³ + b³ + c³ - 3abc = - 25. Class 9

Question 1.

If a + b + c = 5 and ab + bc + ca = 10, then prove that a³ + b³ + c³ - 3abc = - 25. Class 9

Solution:

We know that,

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

∴ 5² = a² + b² + c² + 2 (ab + bc + ac)

∴ 25 = a² + b² + c² + 2(10)

∴ a² + b² + c² = 5

Now,

a³ + b³ + c³ - 3abc

= (a + b + c)(a² + b² + c² - ab - bc - ac)

= (a + b + c)[(a² + b² + c² - (ab + bc + ac)]

= 5(5 -10)

= 5(-5)

= -25