2020HSCShort answer

Q11 (e)

4 marks

Question

Solve $z^2+3z+(3-i)=0$ , giving your answer(s) in the form $a+bi$ , where $a$ and $b$ are real.

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	4
Obtains the square root of the discriminant, or equivalent merit	3
Makes progress towards finding the square root of the discriminant	2
Correctly obtains the discriminant, or equivalent merit<br>OR<br>Attempts to use quadratic formula	1

Sample answer:

z^2+3z+(3-i)=0

\Delta=9-4(3-i)=-3+4i

Write $-3+4i=(x+iy)^2$ .

x^2-y^2=-3\quad\text{(real parts)}

x^2+y^2=5\quad\text{(moduli)}

x^2=1\Rightarrow x=\pm1\Rightarrow y=\pm2

-3+4i=(1+2i)^2

z=\frac{-3\pm(1+2i)}2=-1+i,\,-2-i

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Syllabus alignment

Complex Numbers

Arithmetic of complex numbersRepresent and calculate with complex numbers in Cartesian form, including conjugates, division, reciprocals and square roots.
Solving equations with complex numbersUse De Moivre's theorem and solve equations and polynomial problems involving complex and conjugate roots.

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