Question

Rezolvati ecuati: $(x-1)^{2}+ (x-3)^{2} +...+ (x-99)^{2} = (x+2)^{2}+(x+4) ^{2}+...+(x+100)^{2}$.

Answer 1

[tex][(x-1)^2-(x+2)^2]+[(x-3)^2-(x+4)^2]+...+\\ +[(x-99)^2-(x+100)^2]=0\\ (x-1+x+2)(x-1-x-2)+(x-3+x+4)(x-3-x-4)+...+\\ +(x-99+x+100)(x-99-x-100)=0\\ (2x+1)(-1-2)+(2x+1)(-3-4)+...+(2x-1)(-99-100)=0\\ (2x+1)(-1-2-3-4-...-99-100)=0\\ 2x+1=0\\ 2x=-1\\ x=- \frac{1}{2} [/tex]