Question 1
A solution of the equation $z^2=ai$, where $a\in\mathbb{R}$, is $z=-2-2i$.
The other solution is
Not quite. $-8i$ is what you get by taking the negative of the square, $(-2-2i)^2=8i$. The other solution is the negative of the root itself. Squaring $-8i$ gives $(-8i)^2=-64$, so it can’t solve $z^2=8i$. In 2022, 5.6% of Queensland students chose (A).
Not quite. $-2+2i$ is the conjugate of the given root. Conjugate pairs come from equations with real coefficients, and $z^2-8i=0$ has the imaginary coefficient $-8i$. Squaring gives $(-2+2i)^2=4-8i-4=-8i$, the wrong sign. In 2022, 49.9% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2022, 26.5% of Queensland students chose (C).
Not quite. $8i$ is the value of $z^2$ (the right-hand side $ai$ with $a=8$). The question asks for $z$, and $(8i)^2=-64$. In 2022, 17.5% of Queensland students chose (D).
Solid circle: $|w|=2\sqrt2$. Dashed circle: $|w^2|=8$. Each blue arc runs from the positive real axis to its point, so it shows the argument. The outer arc always turns through twice the angle of the inner one.
This is the given root. Its square lands on $8i$, so $a=8$. Turn $w$ until $w^2$ lands on $8i$ again.
Half a turn from the given root, $w=2+2i$ and $w^2$ is back on $8i$. This is option (C).
This is option (B). It sits on the right circle, but doubling its angle gives $\frac{3\pi}{2}$, so its square is $-8i$.
$w^2$ turns through twice the angle of $w$. Keep turning until $w^2$ lands on $8i$.
Square the given root to find $a$.
$$(-2-2i)^2=4+8i+4i^2=8i$$
So $ai=8i$ and $a=8$.
If $z^2=8i$ then $(-z)^2=z^2=8i$ as well. The two square roots of a number are negatives of each other, half a turn apart on the Argand diagram.
$$z^2=8i\quad\Rightarrow\quad z=\pm(-2-2i)$$
Take the negative of the given root and check it.
$$-(-2-2i)=2+2i$$
$$(2+2i)^2=4+8i-4=8i$$
The given root fixes the equation: $(-2-2i)^2=8i$, so $z^2=8i$.
A number and its negative have the same square, so the other solution is the negative of the given root.
$$z=-(-2-2i)=2+2i$$
On your GDC you can also solve $z^2=8i$ directly and get both roots, $2+2i$ and $-2-2i$.
How Queensland students answered in 2022
More students chose (B) than the answer. From the QCAA Specialist Mathematics subject report for 2022. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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