Nonnegative Squares
Idea · Beginner
Algebra
For every real number ,
with equality if and only if
Therefore, whenever an expression can be written as a sum of squares
we immediately know
Moreover,
if and only if
This gives a powerful strategy:
Rewrite the desired inequality as a sum of nonnegative squares.
For example,
because
Equality occurs exactly when
Completing the Square
Hence
Find the minimum value of
Complete the squares:
Therefore, the minimum is
attained at
Prove that
Subtract the right-hand side:
Let and be real numbers. Find the minimum value of
Equality holds when
Thus, the minimum value is
Find all real numbers and such that
Let
Then , so
Since each square is nonnegative, all three squares must be zero. Hence
Therefore,
For positive real numbers , , and , prove that
We rewrite the desired inequality as
Using the difference of two squares,
Since , this becomes
Both terms are squares of real numbers, so the inequality follows.
The property
is valid when is real.
It does not apply to complex numbers. For example,
Also, over the real numbers,
implies
But this is false over the complex numbers. For example,
even though
So whenever you use the nonnegative squares method, make sure the quantities being squared are real.