Exercise 431Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

A quadratic with complex roots

#quadratic#complex roots#conjugate roots

Statement

Solve in $\mathbb{C}$ the equation $z^2-2z+5=0$ .

Hint

$\Delta<0$ . Use the square root of the negative discriminant.

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Discriminant. $\Delta=4-20=-16=(4i)^2$ .

Roots.
$z=\frac{2\pm4i}{2}=1\pm2i.$

Conclusion. $z\in\{1+2i,1-2i\}$ .

Remark. The roots are conjugate because the coefficients are real.