Complex numbers: Roots and polynomials
Solving quadratic equations in ℂ
Let us return to where we began this chapter with: we have the complex unit \(\mathrm{i}\) introduced to be able to solve the equation \(z^2=-1\). But then you can solve any quadratic equation, even if the discriminant is negative. First, a few examples to illustrate:
We solve the quadratic equation by completing the square: \[\begin{aligned}z^2+6\, z+18=0 &\qquad \color{blue}{\text{the given equation}}\\ \\ \left(z+3\right)^2+9=0 &\qquad\color{blue}{\text{completing the square}}\\ \\ (z+3)^2=-9 &\qquad \color{blue}{\text{constant to the right}} \\ \\ z+3=3\,\mathrm{i}\quad \lor\quad z+3=-3\,\mathrm{i} &\qquad \color{blue}{\mathrm{i}^2=-1} \\ \\ z=-3+3\,\mathrm{i}\quad \lor\quad z= -3-3\,\mathrm{i}&\qquad \color{blue}{\text{solutions in standard form}} \end{aligned}\] Above, we have used the logical "or" operator \(\lor\).
So, there are two solutions: \[z=-3+3\,\mathrm{i}\quad\text{or}\quad z=-3-3\,\mathrm{i}\]
So, there are two solutions: \[z=-3+3\,\mathrm{i}\quad\text{or}\quad z=-3-3\,\mathrm{i}\]
The following example shows that the best quadratic equation may also have complex coefficients.
Solve the following quadratic equation in \(\mathbb{C}\) by: \[z^2-6\, \ii\, z-18=0\]
We solve the quadratic equation by completing the square: \[\begin{aligned}z^2-6\, \ii\, z-18=0 &\qquad \color{blue}{\text{the given equation}}\\ \\ \left(z-3\, \ii\right)^2-9=0 &\qquad\color{blue}{\text{completing the square with }\ii^2=-1}\\ \\ (z-3\,\mathrm{i})^2=9 &\qquad\color{blue}{\text{constant to the right}} \\ \\z-3\,\mathrm{i}=3\quad \lor\quad z-3\,\mathrm{i}=-3 &\qquad\color{blue}{\text{taking roots}} \\ \\ z=3+3\,\mathrm{i}\quad \lor\quad z=-3+3\,\mathrm{i} &\qquad\color{blue}{\text{solutions in standard form}}\end{aligned}\] So there are two solutions: \[z=3+3\,\mathrm{i}\quad\text{or}\quad z=-3+3\,\mathrm{i}\]
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