Find both roots in unit polar form with principal argument in (−π,π].

01 / Standard#Your turn

z^2-\sqrt3z+1=0

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the quadratic formula over the complex numbers.

Hint 2

Both roots have modulus one.

Worked solution

The discriminant is −1.
$z=\frac{\sqrt3\pm i}{2}$
Identify cosine and sine coordinates on the unit circle.
$z=\cos\left(\pm\frac\pi6\right)+i\sin\left(\pm\frac\pi6\right)$

Arguments π/6 and −π/6, both of modulus one.

Checks and common pitfalls: The ± sign changes the sine term as well as the angle.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The discriminant is −1.
$z=\frac{\sqrt3\pm i}{2}$
Identify cosine and sine coordinates on the unit circle.
$z=\cos\left(\pm\frac\pi6\right)+i\sin\left(\pm\frac\pi6\right)$

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

2026 JM02