Quadratic equation in C

Problem

Solve in $\mathbb{C}$ the equation $x^2 - 2x + 10 = 0$.

Solution

Apply the quadratic formula, interpreting the root of the negative discriminant in $\mathbb{C}$: $\Delta = (-2)^2 - 4\cdot 1\cdot 10 = 4 - 40 = -36, \qquad \sqrt{\Delta} = 6i.$ $x = \frac{2 \pm 6i}{2} = \boxed{1 \pm 3i}.$ The two solutions are complex conjugates, as always happens for an equation with real coefficients when $\Delta<0$.

Links

Topics: Complex numbers
Concepts: Cartesian form · Cartesian plane · Fundamental theorem of algebra
Skills: Solving equations · Using formulae
Exercise type: Solving an equation