$ 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 +

["Understanding Complex Roots and Their Magnitudes: A Breakdown of $ u = \frac{-64 \pm \sqrt{4096 + 576}}{2} $", "When solving the quadratic expression\n$$\nu = \frac{-64 \pm \sqrt{4096 + 576}}{2},\n$$\na common next step is simplifying the square root. We compute:\n$$\n4096 + 576 = 4672.\n$$\nNow, simplify $ \sqrt{4672} $. Factor out the largest perfect square:\n$$\n4672 \div 16 = 292 \quad \ ext{so} \quad 4672 = 16 \cdot 292.\n$$\nThen,\n$$\n\sqrt{4672} = \sqrt{16 \cdot 292} = 4\sqrt{292}.\n$$\nBut $ 292 = 4 \cdot 73 $, so\n$$\n\sqrt{292} = \sqrt{4 \cdot 73} = 2\sqrt{73},\n$$\nand thus\n$$\n\sqrt{4672} = 4 \cdot 2\sqrt{73} = 8\sqrt{73}.\n$$\nTherefore,\n$$\nu = \frac{-64 \pm 8\sqrt{73}}{2} = -32 \pm 4\sqrt{73}.\n$$", "This yields two complex values only if $ 4\sqrt{73} > 32 $, which is true since $ \sqrt{73} \approx 8.54 $, so $ 4\sqrt{73} \approx 34.16 $. Hence, $ u $ is real and distinct. However, a deeper mathematical insight reveals how to analyze $ |z|^2 + |w|^2 $ when $ z + w = 2 + 4i $ and $ zw = 13 - 2i $ — key context often used in quadratic roots.", "---", "### Rich Information from Compute Magnitudes", "Given $ z + w = 2 + 4i $, $ zw = 13 - 2i $, we analyze $ |z|^2 + |w|^2 $. Recall the identity for complex numbers:\n$$\n|z|^2 + |w|^2 = |z + w|^2 + |z - w|^2 - 2|z \overline{w}|,\n$$\nbut this is not standardly helpful directly. A more reliable formula combines magnitude and product:\n$$\n|z|^2 + |w|^2 = \ ext{Re}(z \overline{w}) + |z|^2 + |w|^2 - |z|^2 - |w|^2 + \ ext{adjustments} \quad \ ext{(circular)}.\n$$", "Instead, use the identity:\n$$\n|z|^2 + |w|^2 = \frac{1}{2} \left( |z + w|^2 + |z - w|^2 \right) - \frac{1}{2} \cdot 2\left| z \overline{w} - \overline{z} w \right| \quad \ ext{(not standard)}.\n$$", "Better:\nFrom complex number algebra,\n$$\n|z|^2 + |w|^2 = (z + w)(\overline{z} + \overline{w}) - 2,\ ext{Re}(z \overline{w}) = |z + w|^2 - 2,\ ext{Re}(z \overline{w}).\n$$\nWe already have $ z + w = 2 + 4i $, so\n$$\n|z + w|^2 = 2^2 + 4^2 = 4 + 16 = 20.\n$$", "Now compute $ \ ext{Re}(z \overline{w}) $. Note:\n$$\nz \overline{w} + \overline{z} w = 2,\ ext{Re}(z \overline{w}) = (z + w)(\overline{z} + \overline{w}) - |z|^2 - |w|^2.\n$$\nBut $ (z + w)(\overline{z} + \overline{w}) = |z|^2 + |w|^2 + z\overline{w} + \overline{z}w = S + 2,\ ext{Re}(z\overline{w}) $, with $ S = |z|^2 + |w|^2 $.\nAlso $ zw = 13 - 2i $, so $ \overline{zw} = 13 + 2i $, and\n$$\nz \overline{w} \cdot \overline{z} w = |z|^2 |w|^2 = |zw|^2 = |13 - 2i|^2 = 13^2 + (-2)^2 = 169 + 4 = 173.\n$$\nLet $ a = |z|^2 $, $ b = |w|^2 $. Then $ ab = 173 $, and $ S = a + b $.\nAlso, $ z \overline{w} \cdot \overline{z} w = ab = 173 $, but $ z \overline{w} + \overline{z} w = 2,\ ext{Re}(z \overline{w}) $. Let $ x = \ ext{Re}(z \overline{w}) $.\nThen $ z \overline{w} = x + iy $, $ \overline{z} w = x - iy $, so sum is $ 2x $.\nBut:\n$$\nz \overline{w} + \overline{z} w = (z + w)(\overline{z} + \overline{w}) - |z|^2 - |w|^2 = 20 - S.\n$$\nSo\n$$\n2x = 20 - S \quad \Rightarrow \quad x = 10 - \frac{S}{2}.\n$$\nNow recall:\n$$\n|z|^2 |w|^2 = ab = 173, \quad S = a + b.\n$$\nSo $ a $ and $ b $ are roots of:\n$$\nt^2 - St + 173 = 0.\n$$\nBut $ x^2 \le ab = 173 $ by AM-GM:\n$$\n\left( \frac{a + b}{2} \right)^2 \ge ab \Rightarrow \left( \frac{S}{2} \right)^2 \ge 173 \Rightarrow S^2 \ge 4 \cdot 173 = 692.\n$$\nThus $ S \ge \sqrt{692} \approx 26.306 $. Since earlier $ S = 20 - 2x $, and $ x $ real, $ S $ must be real. But we already know numerically $ S = |z|^2 + |w|^2 = (-32 + 4\sqrt{73})^2 + (-32 - 4\sqrt{73})^2 $.\nCompute:\n$$\n(-32 \pm 4\sqrt{73})^2 = 1024 \mp 256\sqrt{73} + 16 \cdot 73 = 1024 \mp 256\sqrt{73} + 1168 = 2192 \mp 256\sqrt{73}.\n$$\nSum: $ S = (2192 - 256\sqrt{73}) + (2192 + 256\sqrt{73}) = 4384 $.\nWait — contradiction? No: both values are $ |z|^2 $ and $ |w|^2 $? No: actually,\nLet $ z = -32 + 4\sqrt{73} $, then $ |z|^2 = (-32)^2 + (4\sqrt{73})^2 = 1024 + 16 \cdot 73 = 1024 + 1168 = 2192 $.\nSimilarly, $ |w|^2 = 2192 $. So $ a = b = 2192 $? No — that would imply $ S = 4384 $, but $ z + w = 2 + 4i $, so $ |z + w|^2 = 20 $, but $ |z + w|^2 = |z|^2 + |w|^2 + 2\ ext{Re}(z\overline{w}) = S + 2x $.\nEarlier, $ 2x = 20 - S \Rightarrow S + (20 - S) = 20 $, consistent.\nBut from magnitude: $ |z|^2 = (-32 + 4\sqrt{73})^2 $. Compute:\n$$\n4\sqrt{73} \approx 4 \cdot 8.544 = 34.176, \quad -32 + 34.176 = 2.176, \quad |z|^2 \approx 4.74, \quad \ ext{but } 2192 \gg 4.74.\n$$\nMistake: $ z = -32 + 4\sqrt{73} $ is not the root — earlier simplification $ \sqrt{4672} = 8\sqrt{73} $ gives $ u = -32 \pm 4\sqrt{73} $, so $ z, w = -32 + 4\sqrt{73} $, but this is a real number! Then $ |z|^2 = z^2 $, since $ z \in \mathbb{R} $. But $ z = -32 + 4\sqrt{73} \approx -32 + 34.176 = 2.176 $, so $ |z|^2 \approx (2.176)^2 \approx 4.74 $. But $ (-32 + 4\sqrt{73})^2 = 1024 - 256\sqrt{73} + 1168 = 2192 - 256\sqrt{73} \approx 2192 - 256 \cdot 8.544 \approx 2192 - 2188.5 = 3.5 $, close.\nBut $ u $ is a root of quadratic — so $ z, w $ are complex in general, but here simplification gave real expression? That suggests discriminant perfect square: $ 4096 + 576 = 4672 $, but $ 4672 $ is not a perfect square. $ \sqrt{4672} \approx 68.35 $, so roots are $ (-64 \pm 68.35)/2 \Rightarrow \approx 2.175 $ and $ -66.175 $.\nThus $ z, w $ are real? Only if discriminant $ \geq 0 $. $ 4672 > 0 $, so roots real. So $ z, w \in \mathbb{R} $. Then $ |z|^2 + |w|^2 = z^2 + w^2 = (z + w)^2 - 2zw = (2 + 4i)^2 - 2(13 - 2i) $. But $ (2 + 4i)^2 = 4 + 16i - 16 = -12 + 16i $, not real — contradiction unless $ zw $ compensates.", "Wait — error: $ z + w = 2 + 4i $, $ zw = 13 - 2i $, so\n$$\nz^2 + w^2 = (z + w)^2 - 2zw = (2 + 4i)^2 - 2(13 - 2i) = (-12 + 16i) - (26 - 4i) = -38 + 20i <br/>\ne S.\n$$\nBut $ |z|^2 + |w|^2 $ for real $ z, w $ is $ z^2 + w^2 $, which is complex — impossible. So $ z, w $ must be complex conjugates? But sum $ 2 + 4i $ not real — so not conjugates. Contradiction.\nResolution: $ z + w = 2 + 4i $, $ zw = 13 - 2i $, so they are roots of $ x^2 - (2 + 4i)x + (13 - 2i) = 0 $, discriminant $ D = 4672 $, not a perfect square, so roots are complex but not conjugates. But $ |z|^2 + |w|^2 $ is still defined: $ |z|^2 = z \overline{z} $, etc.", "Let $ z, w $ be roots. Then\n$$\n|z|^2 + |w|^2 = z\overline{z} + w\overline{w}.\n$$\nBut $ z + w = s = 2 + 4i $, $ zw = p = 13 - 2i $.\nThen $ \overline{z} + \overline{w} = \overline{s} = 2 - 4i $, $ \overline{z}\overline{w} = \overline{p} = 13 + 2i $.\nNow compute:\n$$\n|S|^2 = |z|^2 + |w|^2 = z\overline{z} + w\overline{w}.\n$$\nUse identity:\n$$\n|z + w|^2 + |z - w|^2 = 2(|z|^2 + |w|^2).\n$$\nWe have $ |z + w|^2 = |2 + 4i|^2 = 4 + 16 = 20 $.\nNow compute $ z - w = \sqrt{(z + w)^2 - 4zw} $, but complex square root. Instead:\n$$\n(z - w)^2 = (z + w)^2 - 4zw = (2 + 4i)^2 - 4(13 - 2i) = (-12 + 16i) - (52 - 8i) = -64 + 24i.\n$$\nSo $ z - w $ is a complex number with $ (z - w)^2 = -64 + 24i $. Then\n$$\n|z - w|^2 = |(z - w)^2| = |-64 + 24i| = \sqrt{64^2 + 24^2} = \sqrt{4096 + 576} = \sqrt{4672} = 8\sqrt{73}.\n$$\nThus $ |z - w|^2 = 4672 $.\nThen from identity:\n$$\n|z + w|^2 + |z - w|^2 = 2(|z|^2 + |w|^2) \Rightarrow 20 + 4672 = 2S \Rightarrow 4692 = 2S \Rightarrow S = 2346.\n$$\nSo\n$$\n|z|^2 + |w|^2 = 2346.\n$$", "---", "### Why This Matters Beyond Algebra", "This problem illustrates how complex roots, even when simplifying yields "nice" radicals, encode deep geometric and algebraic structure. The magnitude sum depends not just on sum and product, but on the full complex space spanned by the roots. Techniques from complex analysis and polynomial theory — such as $ |z|^2 = z \overline{z} $, and identities involving conjugates — are essential for precise computation.", "While real roots may seem simpler, complex roots enrich interpretations, especially in engineering and physics where phase and magnitude interplay. Accepting that $ u = -32 \pm 4\sqrt{73} $ is valid (though not elementary) reminds us that even seemingly messy forms can represent precise quantities, and simplification via identities unlocks deeper insight.", "In summary, $ \sqrt{4672} = 8\sqrt{73} $ leads to $ u = -32 \pm 4\sqrt{73} $, and $|z|^2 + |w|^2 = 2346$ via the identity $|z + w|^2 + |z - w|^2 = 2(|z|^2 + |w|^2)$, with $|z - w|^2 = |(z - w)^2| = | -64 + 24i | = \sqrt{4672}$.", "This approach transforms algebraic computation into geometric insight — a hallmark of advanced mathematical reasoning."]