Question

if a+b+c= 12 and a^2 +b^2+c^2= 64 , find the value of ab + bc + ca

Answer 1

Hey !

Solution :

Given :

a + b + c = 12

a² + b² + c² = 64

To find :

ab + bc + ca = ?

Proof :

Consider the identity ( a + b + c )². The solution for that identity is :

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

This can be simplified and written as :

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

Now the values for some variables are given in the question.

( a + b + c ) = 12

( a² + b² + c² ) = 64

 Hence substituting them in the above identity we get,

( 12 )² = ( 64 ) + 2 ( ab + bc + ac )

144 = 64 + 2 ( ab + bc + ac )

144 - 64 = 2 ( ab + bc + ac )

80 = 2 ( ab + bc + ac )

Now 2 in the Right hand side of the equation is in multiplication. Hence while transposing it to the Left hand side it changes into division. Hence we get,

80 / 2 = ( ab + bc + ac )

=> ( ab + bc + ac ) = 40

Therefore the required value is 40.

Hope helped :-)