Problem
ALG-B1-M03-P022 A System with No Hidden Alternatives
#22
★★★★☆ Level 4 of 5
Real numbers \(x,y\) satisfy \(x^2+y^2=2x+4y-5\). Prove that \(x=1\), \(y=2\).
Move everything to the left and complete squares.
Rewrite as \(x^2-2x+y^2-4y+5=0\).
This is \((x-1)^2+(y-2)^2=0\). A sum of squares is zero only when \(x=1\), \(y=2\).
Completing the square is a method for equations too.