Let x and y be real numbers such that x^2 + y^2 = 4 (x + y). Find the largest possible value of x.

The largest possible value of x is 2 + √8

Step-by-step explanation:

Given;

x² + y² = 4 (x + y)

This is an equation of circle

x² + y² = 4x + 4y

x² - 4x + y² - 4y = 0

complete the square by taking half of coefficient of x and y, then add the squares to both sides;

x² - 4x + (-2) ² + y² - 4y + (-2) ² = (-2) ² + (-2) ²

factorize

(x - 2) ² + (y - 2) ² = √8

This circle has its center at (2, 2) with a radius of √8

The largest x-value occurs at the right end of the circle = x value of the center plus the radius = 2 + √8