Question: Let $ z $ and $ w $ be complex numbers such that $ z + w = 2 + 4i $ and $ z \cdot w = 13 - 2i $. Find $ |z|^2 + |w|^2 $.

["Title: Solving Complex Numbers: Find $ |z|^2 + |w|^2 $ Given Their Sum and Product", "When complex numbers $ z $ and $ w $ satisfy $ z + w = 2 + 4i $ and $ z \cdot w = 13 - 2i $, finding $ |z|^2 + |w|^2 $ becomes a natural algebraic exploration. While $ z $ and $ w $ are complex, there’s a elegant way to compute this sum of squared magnitudes using identities that apply to all complex numbers — no need to find $ z $ and $ w $ explicitly.", "Let’s begin by recalling a key identity from complex numbers:", "[\n|z|^2 + |w|^2 = |z + w|^2 - 2,\ ext{Re}(z\overline{w}) + 2,\ ext{Re}(z\overline{w}) \quad \ ext{(not directly helpful)}\n]", "However, a more useful identity combines both sum and product:", "Using the identity:", "[\n|z|^2 + |w|^2 = |z + w|^2 - 2,\ ext{Re}(z \overline{w}) \quad \ ext{? No — this is incorrect.}\n]", "Wait — correct approach: Recall that for complex numbers,", "[\n|z|^2 + |w|^2 = (z + w)(\overline{z + w}) - 2,\ ext{Re}(z \overline{w}) \quad \ ext{still messy.}\n]", "Better idea: Use algebraic identities involving $ |z|^2 + |w|^2 $ directly.", "We know:", "[\n|z|^2 + |w|^2 = z \overline{z} + w \overline{w}\n]", "But we lack conjugates. However, here’s a powerful trick: consider $ (z + w)^2 $ and $ (z - w)^2 $, but since $ z $ and $ w $ are complex, consider expressing $ |z|^2 + |w|^2 $ in terms of $ z + w $ and $ zw $.", "Instead, use the identity:", "[\n|z|^2 + |w|^2 = \frac{1}{2} \left( |z + w|^2 + |z - w|^2 \right)\n]", "But we don’t know $ z - w $. So go back to basics.", "Let $ s = z + w = 2 + 4i $, $ p = zw = 13 - 2i $.", "Now, recall that $ |z|^2 + |w|^2 = z\overline{z} + w\overline{w} $, but again, conjugates are tricky.", "Instead, consider the identity:", "[\n|z|^2 + |w|^2 = (z + w)(\overline{z + w}) - 2,\ ext{Re}(z\overline{w}) \quad \ ext{still not helpful.}\n]", "Wait — here’s the right identity:", "For any complex numbers $ z, w $:", "[\n|z|^2 + |w|^2 = |z + w|^2 - 2,\ ext{Re}(z \overline{w}) + 2,\ ext{Re}(z \overline{w}) \quad \ ext{no.}\n]", "Actually, $ |z + w|^2 = |z|^2 + |w|^2 + 2,\ ext{Re}(z\overline{w}) $, which gives:", "[\n|z|^2 + |w|^2 = |z + w|^2 - 2,\ ext{Re}(z\overline{w})\n]", "But we don’t know $ z\overline{w} $. So we need another route.", "Let’s suppose $ z $ and $ w $ are roots of the quadratic equation:", "[\nx^2 - (z + w)x + zw = 0 \Rightarrow x^2 - (2 + 4i)x + (13 - 2i) = 0\n]", "We don’t need to solve this, but we can use the discriminant to find $ |z|^2 + |w|^2 $ indirectly.", "Let $ D = (2 + 4i)^2 - 4(13 - 2i) $", "Compute:", "[\n(2 + 4i)^2 = 4 + 16i + 16i^2 = 4 + 16i - 16 = -12 + 16i\n]", "[\n4(13 - 2i) = 52 - 8i\n]", "[\nD = (-12 + 16i) - (52 - 8i) = -64 + 24i\n]", "Now, the roots are:", "[\nz, w = \frac{2 + 4i \pm \sqrt{-64 + 24i}}{2} = 1 + 2i \pm \frac{\sqrt{-64 + 24i}}{2}\n]", "Let $ \sqrt{-64 + 24i} = a + bi $, solve:", "[\n(a + bi)^2 = a^2 - b^2 + 2abi = -64 + 24i\n]", "So:", "[\na^2 - b^2 = -64, \quad 2ab = 24 \Rightarrow ab = 12\n]", "From $ ab = 12 $, $ b = 12/a $. Substitute:", "[\na^2 - \left(\frac{144}{a^2}\right) = -64\n\Rightarrow a^4 + 64a^2 - 144 = 0\n]", "Let $ u = a^2 $: $ u^2 + 64u - 144 = 0 $", "Solutions:", "[\nu = \frac{-64 \pm \sqrt{4096 + 576}}{2} = \frac{-64 \pm \sqrt{4672}}{2}\n]", "This is messy — not helpful for $ |z|^2 + |w|^2 $. So return to identity.", "Recall:", "For complex $ z, w $, we have the identity:", "[\n|z|^2 + |w|^2 = |z + w|^2 - 2,\ ext{Re}(z \overline{w}) \quad \ ext{no — still stuck}\n]", "Wait — better idea: use", "[\n|z|^2 + |w|^2 = (z + w)(\overline{z + w}) - 2,\ ext{Re}(z \overline{w}) \quad \ ext{still not identity}\n]", "Actually, there’s a known identity:", "[\n|z|^2 + |w|^2 = |z + w|^2 - 2,\ ext{Re}(z \overline{w}) \quad \ ext{but } \ ext{Re}(z \overline{w}) \ ext{ is not traceable directly}\n]", "But here’s the correct and elegant method:", "We know:", "[\n(z + w)(\overline{z + w}) = |z + w|^2 = |2 + 4i|^2 = 2^2 + 4^2 = 4 + 16 = 20\n]", "But $ |z + w|^2 = |z|^2 + |w|^2 + 2,\ ext{Re}(z \overline{w}) $", "Still involves conjugate.", "Wait — here’s the key: use the identity:", "[\n|z|^2 + |w|^2 = \frac{1}{2} \left( |z + w|^2 + |z - w|^2 \right)\n]", "But we don’t know $ z - w $.", "Alternative idea: use algebraic manipulation with sum and product.", "Let $ s = z + w = 2 + 4i $, $ p = zw = 13 - 2i $", "Then $ |z|^2 + |w|^2 = z\overline{z} + w\overline{w} $", "But consider $ |z + w|^2 = |z|^2 + |w|^2 + 2,\ ext{Re}(z\overline{w}) $", "Still not helpful.", "Wait — consider $ (z + w)(\overline{z} + \overline{w}) = |z|^2 + |w|^2 + z\overline{w} + \overline{z}w = |z|^2 + |w|^2 + 2,\ ext{Re}(z\overline{w}) $", "And $ |z + w|^2 = |2 + 4i|^2 = 20 $", "So:", "[\n|z|^2 + |w|^2 + 2,\ ext{Re}(z\overline{w}) = 20 \quad \ ext{(1)}\n]", "But we need another equation.", "Now consider $ |zw|^2 = |13 - 2i|^2 = 13^2 + (-2)^2 = 169 + 4 = 173 $", "But $ |zw|^2 = |z|^2 |w|^2 $", "Let $ a = |z|^2 $, $ b = |w|^2 $. Then $ ab = 173 $", "From (1): $ a + b + 2,\ ext{Re}(z\overline{w}) = 20 $", "But $ z\overline{w} $ is complex — we need its real part.", "Note: $ z\overline{w} + \overline{z}w = 2,\ ext{Re}(z\overline{w}) $", "But $ z\overline{w} = ? $", "Another idea: consider the discriminant of the quadratic is messy, so instead use:", "Let $ z $ and $ w $ satisfy $ x^2 - (2+4i)x + (13-2i) = 0 $", "The roots are:", "[\nx = \frac{2+4i \pm \sqrt{(2+4i)^2 - 4(13-2i)}}{2} = \frac{2+4i \pm \sqrt{-64 + 24i}}{2}\n]", "Let $ d = \sqrt{-64 + 24i} $. Let $ d = a + bi $, solve:", "$ a^2 - b^2 = -64 $, $ 2ab = 24 \Rightarrow ab = 12 $", "Then $ a^2 - b^2 = -64 $, $ a^2 + b^2 = ? $", "Let $ u = a^2 $, $ v = b^2 $. Then $ u - v = -64 $, $ u + v = ? $", "Also $ ab = 12 \Rightarrow a^2 b^2 = 144 \Rightarrow u v = 144 $", "So:", "$ u - v = -64 $, $ uv = 144 $", "Then $ u, v $ roots of $ t^2 + 64t + 144 = 0 $", "Discriminant: $ 4096 - 576 = 3520 = 64 \cdot 55 $", "So $ t = \frac{-64 \pm 8\sqrt{55}}{2} = -32 \pm 4\sqrt{55} $", "Then $ a^2 = -32 + 4\sqrt{55} $ (positive?), but $ \sqrt{55} \approx 7.4 $, so $ 47.4 = 29.6 $, $ -32 + 29.6 = -2.4 < 0 $ — invalid.", "Wait — $ a^2 = u $ must be positive. $ -32 + 4\sqrt{55} \approx -32 + 29.6 = -2.4 < 0 $ — impossible.", "Mistake: $ u - v = -64 $, $ uv = 144 $, so $ u, v $ solutions to:", "$ u = \frac{ -64 \pm \sqrt{64^2 + 4 \cdot 144} }{2} = \frac{ -64 \pm \sqrt{4096 + 576} }{2} = \frac{ -64 \pm \sqrt{4672} }{2} $", "$ \sqrt{4672} = \sqrt{16 \cdot 292} = 4\sqrt{292} = 4\sqrt{4 \cdot 73} = 8\sqrt{73} \approx 8 \cdot 8.54 = 68.32 $", "So $ u = \frac{ -64 \pm 68.32 }{2} $", "So $ u \approx \frac{4.32}{2} = 2.16 $, $ v \approx \frac{-132.32}{2} = -66.16 $ — negative, impossible.", "Wait — $ u - v = -64 $, so $ u = v - 64 $, then $ (v - 64)v = 144 \Rightarrow v^2 - 64v - 144 = 0 $", "Discriminant: $ 4096 + 576 = 4672 $, yes.", "$ v = \frac{64 \pm \sqrt{4672}}{2} = 32 \pm 2\sqrt{4672/4} = 32 \pm 2\sqrt{1168} $", "Better: $ \sqrt{4672} = \sqrt{16 \cdot 292} = 4\sqrt{292} = 4\sqrt{4 \cdot 73} = 8\sqrt{73} $", "So $ u = \frac{ -64 \pm 8\sqrt{73} }{2} = -32 \pm 4\sqrt{73} $", "$ \sqrt{73} \approx 8.54 $, so $ 48.54 = 34.16 $, so $ u = -32 + 34.16 = 2.16 > 0 $, $ v = -32 - 34.16 = -66.16 < 0 $", "But $ u = a^2 $, $ v = b^2 $, so one must be positive — but $ v < 0 $, so only $ u $ is valid? No — actually $ u - v = -64 $, and $ uv = 144 $, so if $ u = -32 + 4\sqrt{73} \approx 2.16 $, $ v = -32 - 4\sqrt{73} \approx -66.16 $, then $ u - v = 2.16 - (-66.16) = 68.32 <br/>\ne -64 $", "I see: the equation is $ u - v = -64 $, not $ v - u = 64 $. So $ u = v - 64 $, so $ (v - 64)v = 144 \Rightarrow v^2 - 64v - 144 = 0 $", "Then $ v = \frac{64 \pm \sqrt{4096 + 576}}{2} = \frac{64 \pm \sqrt{4672}}{2} = 32 \pm 2\sqrt{1168} $", "$ \sqrt{1168} = \sqrt{16 \cdot 73} = 4\sqrt{73} \approx 48.544 = 34.176 $", "So $ v = 32 \pm 34.176 $, so $ v \approx 66.176 $ or $ v \approx -2.176 $", "Take $ v = 66.176 > 0 $, so $ b^2 = 66.176 $, $ a^2 = v - 64 = 2.176 $", "So $ |z|^2 = a^2 = 2.176 $, $ |w|^2 = b^2 = 66.176 $, sum $ \approx 68.352 $", "But 173 / (2.176) = ? Wait, $ |zw|^2 = |z|^2 |w|^2 = ab = 173 $", "$ 2.176 * 66.176 \approx 144 $, yes.", "But 173 / 2.176 ≈ 79.5 — inconsistency.", "No — $ |zw|^2 = (|z||w|)^2 = |z|^2 |w|^2 = ab = 173 $", "But if $ |z|^2 = a^2 = u \approx 2.176 $, $ |w|^2 = v \approx 66.176 $, product ≈ 144 ≠ 173 — contradiction.", "Ah, mistake: in $ u = a^2 $, $ v = b^2 $, but $ u - v = -64 $, and $ uv = 144 $, but $ |zw|^2 = u v = 144 $, but given $ |zw|^2 = |13-2i|^2 = 169 + 4 = 173 $, contradiction.", "Wait — error in discriminant.", "$ d^2 = -64 + 24i $, let $ d = a+bi $", "$ a^2 - b^2 = -64 $, $ 2ab = 24 \Rightarrow ab=12 $", "Then $ (a^2 - b^2)^2 = (-64)^2 = 4096 $, $ (a^2 + b^2)^2 = (a^2 - b^2)^2 + 4a^2b^2 = 4096 + 4144 = 4096 + 576 = 4672 $", "So $ a^2 + b^2 = \sqrt{4672} = 8\sqrt{73} $"]