Mathematics - Complex Number Question with Solution | TestHub

${\mathrm{z}}^2+\mathrm{az}+\mathrm{b}=0;{\mathrm{z}}_1+{\mathrm{z}}_2=-\mathrm{a}\&{\mathrm{z}}_1{\mathrm{z}}_2=\mathrm{b}$ $0,\mathrm{z},{\mathrm{z}}_2$ form an equilateral $\Delta$ $\therefore 0^2+{\mathrm{z}}_1{}^2+{\mathrm{z}}_2{}^2=0.{\mathrm{z}}_1+{\mathrm{z}}_1\cdot {\mathrm{z}}_2+{\mathrm{z}}_2\cdot 0$ (for equation $\Delta ,{\mathrm{z}}_1{}^2+{\mathrm{z}}_2{}^2+{\mathrm{z}}_3{}^2={\mathrm{z}}_1{\mathrm{z}}_2+{\mathrm{z}}_2{\mathrm{z}}_3+{\mathrm{z}}_3{\mathrm{z}}_1$ ) $z_1^2+z_2^2=z_1{z}_2$ or ${\left(z_1+z_2\right)}^2=3z_1{z}_2$ $\therefore {\mathrm{a}}^2=3\mathrm{b}\text{. }$