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Home » \binom84 \cdot \binom122 - Dyverse

\binom84 \cdot \binom122 - Dyverse

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Let $ y = x - 1 - rac{1}{x} $ â not helpful. Try numerical values. From $ z + w = 2 + 4i $, $ zw = 13 - 2i $, $ z $ and $ w $ are roots of $ t^2 - (2 + 4i)t + (13 - 2i) = 0 $. Compute discriminant $ D = (2 + 4i)^2 - 4(13 - 2i) = 4 + 16i - 16 - 52 + 8i = -64 + 24i $. Let $ \sqrt{D} = a + bi $, then $ (a + bi)^2 = a^2 - b^2 + 2abi = -64 + 24i $. So:

📌 $ a^2 - b^2 = -64 $,

📌 $ 2ab = 24 \Rightarrow ab = 12 $.

📌 Solve: $ b = 12/a $, substitute: $ a^2 - (144/a^2) = -64 $. Multiply by $ a^2 $: $ a^4 + 64a^2 - 144 = 0 $. Let $ u = a^2 $: $ u^2 + 64u - 144 = 0 $.

📌 $ u = rac{-64 \pm \sqrt{4096 + 576}}{2} = rac{-64 \pm \sqrt{4672}}{2} $. Not real? Wait, $ \sqrt{4672} = \sqrt{16 \cdot 292} = 4\sqrt{292} = 4\sqrt{4 \cdot 73} = 8\sqrt{73} $. So $ u = rac{-64 \pm 8\sqrt{73}}{2} = -32 \pm 4\sqrt{73} $. Take positive root: $ a^2 = -32 + 4\sqrt{73} $, messy. Instead, accept that $ |z|^2 + |w|^2 = |z + w|^2 + |z - w|^2 - 2|z \overline{w}| $, but no. Final correct approach: $ |z|^2 + |w|^2 = (z + w)(\overline{z} + \overline{w}) - 2 ext{Re}(z \overline{w}) = |z + w|^2 - 2 ext{Re}(z \overline{w}) $. But $ z \overline{w} + \overline{z} w = 2 ext{Re}(z \overline{w}) $. Also, $ (z + w)(\overline{z} + \overline{w}) = |z|^2 + |w|^2 + z \overline{w} + \overline{z} w = S + 2 ext{Re}(z \overline{w}) $. From $ z + w = 2 + 4i $, $ zw = 13 - 2i $, consider $ |z + w|^2 = 20 $, $ |zw|^2 = 173 $. Use identity: $ |z|^2 + |w|^2 = \sqrt{ |z + w|^4 + |z - w|^2 |z + w|^2 - 2|z|^2 |w|^2 } $ â too complex. Given time, assume a simpler path: From $ z + w = 2 + 4i $, $ zw = 13 - 2i $, compute $ |z|^2 + |w|^2 = |z + w|^2 + |z - w|^2 - 2|z \overline{w}| $. Not working. Use $ |z|^2 + |w|^2 = (z + w)(\overline{z} + \overline{w}) - 2 ext{Re}(z \overline{w}) = 20 - 2 ext{Re}(z \overline{w}) $. Now, $ z \overline{w} + \overline{z} w = 2 ext{Re}(z \overline{w}) $. Let $ S = |z|^2 + |w|^2 $, $ P = |z|^2 |w|^2 = 173 $. Also, $ (z + w)(\overline{z} + \overline{w}) = S + z \overline{w} + \overline{z} w

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