Question

if a+b+c=5 and ab+bc+ca=10 then prove a power 3+ b power 3+c power 3=-25

Answer 1

Identity used
A³+b3+c3- 3abc = (a+b+c)*(a2+b2+c2-ab-bc-ca) [It is an identity]

a+b+c= 5 (given).

ab+ bc + ca = 10 (given)

Now concentrate a bit, if you look carefully at the identity, we have got 2 values, we need to find a2+b2+c2.

(a+b+c)*[a2+b2+c2- ( ab+bc+ca) ] ( -ab-bc-ca can be written as - (ab+bc+ca, u must know this ofc )

Putting the values we get-

5* ( a2+b2+c2- 10)

We need to find a2+b2+c2, AND with values we have been given. If we think a little, we get this :-

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

52 = a2+b2+c2+ 2( ab+bc+ca) [ Taking 2 as common ]

25= a2+b2+c2 + 2(10)

25 = a2+b2+c2 + 20

Therefore, a2+b2+c2= 5

We got all values! letz put them in the equation we got :-

5* ( 5-10)

= 5* (-5)

= -25.

Proved!