Quadratic Equation/Examples
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: $x^2 + 4 = 0$
The quadratic equation:
- $x^2 + 4 = 0$
has the wholly imaginary roots:
- $x = \pm 2 i$
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 $b, c \in \R$ be real.
Let the quadratic equation:
- $z^2 + b z + c = 0$
have the root:
- $z = 2 + i$
Then the other root is:
- $z = 2 - i$
and it follows that:
| \(\ds b\) | \(=\) | \(\ds -4\) | ||||||||||||
| \(\ds c\) | \(=\) | \(\ds 5\) |
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:
- $z = \begin{cases} \cos \dfrac {2 \pi} 9 \pm i \sin \dfrac {2 \pi} 9 \\ \cos \dfrac {4 \pi} 9 \pm i \sin \dfrac {4 \pi} 9 \\ \cos \dfrac {8 \pi} 9 \pm i \sin \dfrac {8 \pi} 9 \end{cases}$
Example: $3 x^2 + 24 x + 9 = 0$
The quadratic equation:
- $3 x^2 + 24 x + 9 = 0$
has the roots:
- $x = -4 \pm \sqrt {13}$