Problem
ALG-B1-M06-P009 No real solutions
#9
★★☆☆☆ Level 2 of 5
Prove that the equation \(x^2+4y^2-4x-8y+13=0\) has no real solutions.
Complete the squares, taking into account the coefficient \(4\) before \(y^2\).
The left-hand side equals \((x-2)^2+4(y-1)^2+5\). This is always at least \(5\), so it cannot be zero.
Shows that completing the square proves not only a minimum, but also impossibility.