Solution: Let $w = z^2$, so $w^2 + w + 1 = 0$. Solving gives $w = \frac{-1 \pm \sqrt{-3}}{2} = e^{\pm 2\pi i/3}$. Thus, $z^2 = e^{\pm 2\pi i/3}$, so $z = e^{\pm \pi i/3 + \pi i k/2}$ for $k = 0, 1$. The roots have angles $\pi/3, 5\pi/3, 7\pi/3, 11\pi/3$, but

["Solving $z^2 = w$ When $w^2 + w + 1 = 0$: A Clean Approach Using Roots of Unity", "In complex number theory, solving equations involving roots and symmetry often reveals elegant geometric interpretations. A compelling example involves the equation:", "$$\nw^2 + w + 1 = 0\n$$", "Solving this quadratic gives:", "$$\nw = \frac{-1 \pm \sqrt{-3}}{2} = e^{\pm \frac{2\pi i}{3}}\n$$", "These roots lie on the unit circle at angles $ \frac{2\pi}{3} $ and $ \frac{4\pi}{3} $ (since $-\frac{2\pi}{3} \equiv \frac{4\pi}{3} \mod 2\pi$).", "So we now solve:", "$$\nz^2 = e^{\pm \frac{2\pi i}{3}}\n$$", "This requires finding all complex square roots of these complex numbers. Recall that to take the square root of $ e^{i\ heta} $, the solutions are:", "$$\nz = e^{i\ heta/2 + \pi i k / 2}, \quad k = 0, 1\n$$", "Apply this to both values of $ w $:", "---", "For $ z^2 = e^{\frac{2\pi i}{3}} $, the square roots are:", "$$\nz = e^{\frac{2\pi i}{6} + \pi i k / 2} = e^{\pi i / 3 + \pi i k / 2}, \quad k = 0, 1\n$$", "- $ k = 0 $: $ z = e^{\pi i / 3} $ → angle $ \frac{\pi}{3} $\n- $ k = 1 $: $ z = e^{\pi i / 3 + \pi i / 2} = e^{5\pi i / 3} $ → equivalent to $ -\frac{\pi}{3} $ modulo $ 2\pi $", "But note: $ e^{5\pi i / 3} = \cos\left(\frac{5\pi}{3}\right) + i\sin\left(\frac{5\pi}{3}\right) $, and since angles are typically expressed in $ [0, 2\pi) $, $ \frac{5\pi}{3} $ is valid.", "Wait — let’s be more precise. $ \frac{2\pi i}{3} $ corresponds to angle $ \frac{2\pi}{3} $, so half is $ \frac{\pi}{3} $. Adding $ \pi i k / 2 $ gives:", "- $ k = 0 $: $ \ heta = \frac{\pi}{3} $\n- $ k = 1 $: $ \ heta = \frac{\pi}{3} + \frac{\pi}{2} = \frac{5\pi}{6} $", "Ah! Correction: $ \pi i k / 2 $ adds $ \pi/2 $ radians per step. So:", "- $ z = e^{\frac{2\pi i}{3} / 2} \cdot e^{\pi i k} = e^{\pi i / 3} \cdot e^{\pi i k} = e^{\pi i (1/3 + k)} $, $ k = 0, 1 $", "So:", "- $ k = 0 $: $ z = e^{\pi i / 3} = \ ext{angle } \frac{\pi}{3} $ ($ 60^\circ $)\n- $ k = 1 $: $ z = e^{\pi i / 3 + \pi i} = e^{4\pi i / 3} $ ($ 240^\circ $)", "Now for $ z^2 = e^{-\frac{2\pi i}{3}} = e^{\frac{4\pi i}{3}} $ (since $ -2\pi/3 \equiv 4\pi/3 \mod 2\pi $), so square roots:", "$$\nz = e^{\frac{4\pi i}{3} / 2 + \pi i k} = e^{\frac{2\pi i}{3} + \pi i k}, \quad k = 0, 1\n$$", "- $ k = 0 $: $ z = e^{2\pi i / 3} = \frac{-1 + i\sqrt{3}}{2} $, angle $ \frac{2\pi}{3} $\n- $ k = 1 $: $ z = e^{2\pi i / 3 + \pi i} = e^{5\pi i / 3} $, angle $ \frac{5\pi}{3} $", "Thus, the four roots are:", "- $ z_1 = e^{\pi i / 3} = \frac{1}{2} + i\frac{\sqrt{3}}{2} $, $ \angle \frac{\pi}{3} $\n- $ z_2 = e^{2\pi i / 3} $, $ \angle \frac{2\pi}{3} $\n- $ z_3 = e^{4\pi i / 3} $, $ \angle \frac{4\pi}{3} $\n- $ z_4 = e^{5\pi i / 3} $, $ \angle \frac{5\pi}{3} $", "All lie on the unit circle, equally spaced every $ \frac{\pi}{3} $, starting from $ \frac{\pi}{3} $. This symmetric placement reflects the sixth roots of unity — in fact, these are the primitive 6th roots of unity.", "---", "### Why This Structure Matters", "The roots of $ z^2 = e^{2\pi i/3} $ are not arbitrary — they lie at symmetrically spaced points on the complex plane, forming a regular quadrilateral (rotated square) centered at the origin. This elegance arises because $ e^{\pm 2\pi i/3} $ are cube roots of unity (with $ w^3 = 1 $), and square roots of cube roots of unity generate sixth roots of unity.", "This pattern is vital in signal processing, control theory, and cryptography, where symmetry and periodicity determine system stability and efficiency.", "---", "### Final Thoughts", "Solving $ z^2 = w $ when $ w $ is a root of unity unveils deeper structure: the roots form a regular polygon on the unit circle. Recognizing $ w = e^{\pm 2\pi i/3} $ as cube roots of unity, and taking their square roots, reveals why the solutions $ z $ are spaced at $ \frac{\pi}{3} $ intervals — a hallmark of 6th roots of unity. This method simplifies complex root-finding by leveraging symmetry and known geometric properties.", "Whether you're analyzing oscillatory systems or designing filters, understanding such roots enables elegant, efficient solutions.", "---", "Keywords: $ z^2 = w $, complex roots, roots of unity, $ w^2 + w + 1 = 0 $, $ e^{\pm 2\pi i/3} $, square roots in complex plane, sixth roots of unity, $ \frac{\pi}{3} $, exponential form, $ e^{i\ heta} $, $ z = e^{\pm \pi i /3 + \pi i k /2} $"]