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: \[\begin{aligned}z^2+4\, z+8=0 &\qquad \color{blue}{\text{the given equation}}\\ \\ \left(z+2\right)^2+4=0 &\qquad\color{blue}{\text{completing the square}}\\ \\ (z+2)^2=-4 &\qquad \color{blue}{\text{constant to the right}} \\ \\ z+2=2\,\mathrm{i}\quad \lor\quad z+2=-2\,\mathrm{i} &\qquad \color{blue}{\mathrm{i}^2=-1} \\ \\ z=-2+2\,\mathrm{i}\quad \lor\quad z= -2-2\,\mathrm{i}&\qquad \color{blue}{\text{solutions in standard form}} \end{aligned}\] Above, we have used the logical "or" operator \(\lor\).
So, there are two solutions: \[z=-2+2\,\mathrm{i}\quad\text{or}\quad z=-2-2\,\mathrm{i}\]
So, there are two solutions: \[z=-2+2\,\mathrm{i}\quad\text{or}\quad z=-2-2\,\mathrm{i}\]
The following example shows that the best quadratic equation may also have complex coefficients.
Solve the following quadratic equation in \(\mathbb{C}\) by: \[z^2+10\, \ii\, z-26=0\]
We solve the quadratic equation by completing the square: \[\begin{aligned}z^2+10\, \ii\, z-26=0 &\qquad \color{blue}{\text{the given equation}}\\ \\ \left(z+5\, \ii\right)^2-1=0 &\qquad\color{blue}{\text{completing the square with }\ii^2=-1}\\ \\ (z+5\,\mathrm{i})^2=1 &\qquad\color{blue}{\text{constant to the right}} \\ \\z+5\,\mathrm{i}=1\quad \lor\quad z+5\,\mathrm{i}=-1 &\qquad\color{blue}{\text{taking roots}} \\ \\ z=1-5\,\mathrm{i}\quad \lor\quad z=-1-5\,\mathrm{i} &\qquad\color{blue}{\text{solutions in standard form}}\end{aligned}\] So there are two solutions: \[z=1-5\,\mathrm{i}\quad\text{or}\quad z=-1-5\,\mathrm{i}\]
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