If a+b+c = 7 and ab+bc+ac = 12, then find the value of a² + b² + c²

Question: If $a + b + c = 7$ and $ab + bc + ac = 12$ then the value of $a^2 + b^2 + c^2$ is
a) 25
b) 49
c) 36
d) 19

Answer: a) 25

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Solution:

Given,

$a + b + c = 7$

$ab + bc + ac = 12$

To find: $a^2 + b^2 + c^2 = ?$

We know,

$a + b + c = 7$

$or, (a + b) = (7 - c)$

And,

$ab + bc + ac = 12$

$or, ab + c (a + b) = 12$

$or, ab = 12 - c(a + b)$

Now,

$a^2 + b^2 + c^2$

$= (a^2 + b^2) + c^2$

$= (a + b)^2 - 2ab + c^2$

$= (7 - c)^2 - 2{12 - c(a + b)} + c^2$

$= 49 - 14c + c^2 - 2{12 - c(7 - c)} + c^2$

$= 49 - 14c + 2c^2 - 2{12 - 7c +c^2}$

$= 49 - 14c + 2c^2 - 24 + 14c - 2c^2$

$= 25 + 0$

$= 25$

Hence, option a) 25 is the correct answer.

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