Prove that the expression x ^ 2 + y ^ 2-2xy + 4x-4y + 5 takes only positive values for any values of its variables

Let us prove that the expression x ^ 2 + y ^ 2 – 2 * x * y + 4 * x – 4 * y + 5 takes only positive values for any values of the variables included in it.

Let’s simplify the expression and get:

Let’s check:

Let x = 1, y = 1, then we substitute the known value into the expression and calculate its value.

x ^ 2 + y ^ 2 – 2 * x * y + 4 * x – 4 * y + 5;

(x ^ 2 – 2 * x * y + y ^ 2) + 4 * (x – y) + 5;

(x – y) ^ 2 + 4 * (x – y) + 5;

This means that for any values of x and y, the expression takes on a positive value.

We get:

x ^ 2 + y ^ 2 – 2 * x * y + 4 * x – 4 * y + 5 = 1 ^ 2 + 1 ^ 2 – 2 * 1 * 1 + 4 * 1 – 4 * 1 + 5 = 1 + 1 – 2 + 4 – 4 + 5 = 2 – 2 + 0 + 5 = 0 + 5 = 5.