Given a+b+c=7 and a^2+b^2+c^2=25 find (a+b) ^2+(a+c) ^2+(b+c) ^2

26.

Step-by-step explanation:

(a + b)^2 + (a+ c)^2 + (b + c)^2 =   a^2 + b^2 - 2 ab + a^2 + c^2 - 2ac + b^2 + c^2 - 2bc

= 2( a^2 + b^2  + c^2 ) - 2( ab + ac + bc)

=  50 - 2(ab + ac + bc) (1).

(a + b + c)^2 = a^2 + ab + ac + ab + b^2 + bc + ac + bc + c^2

= a^2 + b^2 + c^2 + 2(ab + ac + bc)

= 25 + 2(ab + ac + bc)

But (a + b + c)^2 = 7^2 = 49. So:-

49 = 25 + 2(ab + ac + bc)

2(ab + ac + bc) = 49 - 25 = 24.

Substituting for this  in (1) above:

(a + b)^2 + (a + c)^2 + (b + c)^2 =  50 - 24 = 26 (answer).