Question

if a+b+c=0 find bc/2a2+bc +ca/2b2+ca +ab/2c2+ab​

Answer 1

bc/2a²+bc +ca/2b²+ca +ab/2c²+ab​ = 0

Step-by-step explanation:

Given: a+b+c = 0

Find: bc/2a²+bc +ca/2b²+ca +ab/2c²+ab​

Solution:  

2a²+bc = a² + a² + bc

  = a² + a (-b-c) + bc

  = a (a-b) - c(a-b)

  = (a-b) (a-c)

Similarly 2b²+ca = (b-c)(b-a) and 2c²+ab​ = (c-a)(c-b)

Now bc/2a²+bc  +ca/2b²+ca  +ab/2c²+ab​ = bc/(a-b) (a-c) + ca/(b-c)(b-a) + ab/(c-a)(c-b)

= bc(b-c) + ca(c-a) + ab(a-b) / (a-b)(b-c)(c-a)

= b²c - bc² +ac² -a²c +a²b - ab² / (a-b)(b-c)(c-a)

= 0 / (a-b)(b-c)(c-a)

= 0

bc/2a²+bc +ca/2b²+ca +ab/2c²+ab​ = 0