Quadratic Equation/Examples

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Examples of Quadratic Equation

Example: $x^2 + 1 = 0$

The quadratic equation:

$x^2 + 1 = 0$

has no root in the set of real numbers $\R$:

$x = \pm i$

where $i = \sqrt {-1}$ is the imaginary unit.


Example: $z^2 - \paren {3 + i} z + 4 + 3 i = 0$

The quadratic equation in $\C$:

$z^2 - \paren {3 + i} z + 4 + 3 i = 0$

has the solutions:

$z = \begin{cases} 1 + 2 i \\ 2 - i \end{cases}$


Example: $z^2 + \paren {2 i - 3} z + 5 - i = 0$

The quadratic equation in $\C$:

$z^2 + \paren {2 i - 3} z + 5 - i = 0$

has the solutions:

$z = \begin{cases} 2 - 3 i \\ 1 + i \end{cases}$


Example: $z^2 - 4 z + 5 = 0$

Let the quadratic equation:

$z^2 + b z + c = 0$

have the root:

$z = 2 + i$

Let $a, b, c \in \R$ be real.


Then the other root is:

$z = 2 - i$

and it follows that:

\(\displaystyle b\) \(=\) \(\displaystyle -4\)
\(\displaystyle c\) \(=\) \(\displaystyle 5\)


Example: $5 z^2 + 2 z + 10 = 0$

The quadratic equation in $\C$:

$5 z^2 + 2 z + 10 = 0$

has the solutions:

$z = \dfrac {-1 \pm 7 i} 5$


Example: $z^4 + z^2 + 1 = 0$

The quartic equation:

$z^4 + z^2 + 1 = 0$

has the solutions:

$z = \dfrac {\pm 1 \pm i \sqrt 3} 2$


Example: $z^2 + \paren {i - 2} z + \paren {3 - i} = 0$

The quadratic equation in $\C$:

$z^2 + \paren {i - 2} z + \paren {3 - i} = 0$

has the solutions:

$z = \begin{cases} 1 + i \\ 1 - 2 i \end{cases}$


Example: $z^4 - 2 z^2 + 4 = 0$

The quartic equation:

$z^4 - 2 z^2 + 4 = 0$

has the solutions:

$z = \begin{cases} \dfrac {\sqrt 6} 2 + i \dfrac {\sqrt 2} 2 \\ \dfrac {\sqrt 6} 2 - i \dfrac {\sqrt 2} 2 \\ -\dfrac {\sqrt 6} 2 + i \dfrac {\sqrt 2} 2 \\ -\dfrac {\sqrt 6} 2 - i \dfrac {\sqrt 2} 2 \end{cases}$


Example: $z^6 + z^3 + 1 = 0$

The sextic equation:

$z^6 + z^3 + 1 = 0$

has the solutions:

$z = \begin{cases} \cos \dfrac {2 \pi} 9 \pm i \sin \dfrac {2 \pi} 9 \\ \cos \dfrac {4 \pi} 9 \pm i \sin \dfrac {4 \pi} 9 \\ \cos \dfrac {8 \pi} 9 \pm i \sin \dfrac {8 \pi} 9 \end{cases}$