Use the preceding quadratic to find all four roots, with principal arguments.

01 / Standard#Your turn

z^4-\sqrt3z^2+1=0

Official paper · jm02-2026 · 4(a)(ii) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Set w=z².

Hint 2

Each nonzero w has two square roots separated by π.

Worked solution

Apply the quadratic result to z².
$z^2=e^{i\pi/6}\ \text{or}\ e^{-i\pi/6}$
Halve the arguments after allowing an extra full turn, then reduce to the principal interval.
$z=\cos\theta+i\sin\theta,\quad\theta\in\left\{\frac\pi{12},-\frac\pi{12},\frac{11\pi}{12},-\frac{11\pi}{12}\right\}$

Four roots with arguments ±π/12 and ±11π/12.

Checks and common pitfalls: Halving only the two principal arguments would omit two roots.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Apply the quadratic result to z².
$z^2=e^{i\pi/6}\ \text{or}\ e^{-i\pi/6}$
Halve the arguments after allowing an extra full turn, then reduce to the principal interval.
$z=\cos\theta+i\sin\theta,\quad\theta\in\left\{\frac\pi{12},-\frac\pi{12},\frac{11\pi}{12},-\frac{11\pi}{12}\right\}$

Think first. Reveal a hint when the class is ready.

2026 JM02