Quadratic Equation/Examples/z^2 - (3+i)z + 4+3i = 0

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Example of Quadratic Equation

The quadratic equation in $\C$:

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

has the solutions:

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


Proof

From the Quadratic Formula:

\(\ds z^2 - \paren {3 + i} z + 4 + 3 i\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds z\) \(=\) \(\ds \dfrac {-\paren {-\paren {3 + i} } \pm \sqrt {\paren {-\paren {3 + i} }^2 - 4 \times 1 \times \paren {4 + 3 i} } } {2 \times 1}\) Quadratic Formula: $a = 1$, $b = -\paren {3 + i}$, $c = 4 + 3 i$
\(\ds \) \(=\) \(\ds \dfrac {3 + i} 2 \pm \dfrac {\sqrt {\paren {8 + 6 i} - \paren {16 + 12 i} } } 2\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {3 + i} 2 \pm \dfrac {\sqrt {-8 - 6 i} } 2\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {3 + i} 2 \pm \dfrac {\sqrt {\paren {-1} \paren {8 + 6 i} } } 2\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {3 + i} 2 \pm \dfrac {i \sqrt {\paren {8 + 6 i} } } 2\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {3 + i} 2 \pm \dfrac {i \paren {3 + i} } 2\) from above: we have established that $\paren {\pm \paren {3 + i} }^2 = \paren {8 + 6 i}$
\(\ds \) \(=\) \(\ds \dfrac {3 + i} 2 \pm \dfrac {3 i + i^2} 2\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {3 + i} 2 \pm \dfrac {-1 + 3 i} 2\) simplifying
\(\ds \) \(=\) \(\ds \begin{cases} 1 + 2 i \\ 2 - i \end{cases}\) simplifying

$\blacksquare$


Sources