$ 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 } $ — 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

Sun, 03 May 2026 02:29:42 +0000

Content under review