Algebra Help?

Question

If asquared + bsqaured + csquared = 99,
and ab + bc + ac = 35,
find the value of a + b + c.

Answer 1

Consider (a + b + c) ² :-
(a + b + c) (a + b + c)
a ² + ab + ac + b ² +ab + bc + c ² + ac + bc
a ² + b ² + c ² + 2 (ab + bc + ac)
99 + 70
169

(a + b + c) = √169 = 13

Answer 2

(a+b+c)² = a² + b² + c² + 2(ab + bc + ac)
(a+b+c)² = 99 + 2(35)
(a+b+c)² = 99 + 70
(a+b+c)² = 169
a+b+c = squareroot (169)
a+b+c = +- 13

Answer 3

(a+b+c)^2 = a^2 +b^2 +c^2 +2(ab+bc+ca)(this is standard formula)
= 99 + 2(35)
= 99+70
a+b+c =sqr 169
a+b+c =13

Answer 4

(a + b + c)^2 = a^2 + 2a(b + c) + (b + c)^2 =
= a^2 + 2ab + 2ac + b^2 + 2bc + c^2 =
= a^2 + b^2 + c^2 + 2ab + 2bc + 2ac =
= (a^2 + b^2 + c^2) + 2(ab + bc + ac) =
= 99 + 70 = 169.

Take square roots:

a + b + c = +- sqrt(169) = +- 13