We have to prove that a^2 - 4a + b^2 + 10b + 29>=0, for real values of a and b.
a^2 - 4a + b^2 + 10b + 29
=> a^2 - 4a + 4 + b^2 + 10b + 25
=> (a - 2)^2 + (b + 5)^2
The sum of squares of real numbers is always positive.
This proves that a^2 - 4a + b^2 + 10b + 29 >= 0
Alll we need to do is to express the left sides as a sum of postivie numbers.
We'll create perfect squares to the left side:
(a^2 - 4a + 4) + (b^2 + 10b + 25) - 4 - 25 + 29 >=0
We notice that we've added the numbers 4 and 25 to complete the squares. For the inequality to hold, we'll have to subtract these added values.
(a - 2)^2 + (b + 5)^2 - 29 + 29 >= 0
We'll eliminate like terms:
(a - 2)^2 + (b + 5)^2 > =0
Since the squares are always positive, the inequality is verified, for any real values of a and b.