Question

If a+b+c=5 and ab+bc+ca=10 , prove that a^3 +b^3+ c^3 - 3abc = -25.

Answer 1

Hlo friend.. Cutiepie Here..

Here is ur answer :

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

a³ + b³ + c³ - 3abc = ( a + b + c ) ( a² + b² + c² - ( ab + bc + ca) _________ (1)

First we have to find a² + b² + c²

From identity,

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

( 5 )² = a² + b² + c² + 2 ( ab + bc + ca)

25 = a² + b² + c² + 2 ( 10)

25 = a² + b² + c² + 20

a² + b² + c² = 25 - 20

a² + b² + c² = 5

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Putting the value of a² + b² + c² in eqⁿ

a³ + b³ + c³ - 3abc

= ( 5) {5 - ( 10)}

= ( 5)(5-10)

= ( 5)( - 5)

= -25


a³ + b³ + c³ - 3abc = - 25


Hence Proved.

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HOPE IT HELPS YOU..