Quadratic with complex coefficients - Complex Numbers Arithmetic

CAIE P3 2013 November Q9

9

Without using a calculator, use the formula for the solution of a quadratic equation to solve $$( 2 - \mathrm { i } ) z ^ { 2 } + 2 z + 2 + \mathrm { i } = 0$$ Give your answers in the form $a + b \mathrm { i }$.
The complex number $w$ is defined by $w = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }$. In an Argand diagram, the points $A , B$ and $C$ represent the complex numbers $w , w ^ { 3 }$ and $w ^ { * }$ respectively (where $w ^ { * }$ denotes the complex conjugate of $w$ ). Draw the Argand diagram showing the points $A , B$ and $C$, and calculate the area of triangle $A B C$.

CAIE P3 2023 November Q4

4 Solve the quadratic equation $( 3 + \mathrm { i } ) w ^ { 2 } - 2 w + 3 - \mathrm { i } = 0$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.

CAIE P3 2016 November Q9

Solve the equation $( 1 + 2 \mathrm { i } ) w ^ { 2 } + 4 w - ( 1 - 2 \mathrm { i } ) = 0$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $| z - 1 - \mathrm { i } | \leqslant 2$ and $- \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi$.