$ u = rac-64 \pm \sqrt4096 + 5762 = rac-64 \pm \sqrt46722 $. Not real? Wait, $ \sqrt4672 = \sqrt16 \cdot 292 = 4\sqrt292 = 4\sqrt4 \cdot 73 = 8\sqrt73 $. So $ u = rac-64 \pm 8\sqrt732 = -32 \pm 4\sqrt73 $. Take positive root: $ a^2 = -32 + 4\sqrt73 $, messy. Instead, accept that $ |z|^2 + |w|^2 = |z + w|^2 + |z - w|^2 - 2|z \overlinew| $, but no. Final correct approach: $ |z|^2 + |w|^2 = (z + w)(\overlinez + \overlinew) - 2 extRe(z \overlinew) = |z + w|^2 - 2 extRe(z \overlinew) $. But $ z \overlinew + \overlinez w = 2 extRe(z \overlinew) $. Also, $ (z + w)(\overlinez + \overlinew) = |z|^2 + |w|^2 + z \overlinew + \overlinez w = S + 2 extRe(z \overlinew) $. From $ z + w = 2 + 4i $, $ zw = 13 - 2i $, consider $ |z + w|^2 = 20 $, $ |zw|^2 = 173 $. Use identity: $ |z|^2 + |w|^2 = \sqrtz $ 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 \overlinew| $. Not working. Use $ |z|^2 + |w|^2 = (z + w)(\overlinez + \overlinew) - 2 extRe(z \overlinew) = 20 - 2 extRe(z \overlinew) $. Now, $ z \overlinew + \overlinez w = 2 extRe(z \overlinew) $. Let $ S = |z|^2 + |w|^2 $, $ P = |z|^2 |w|^2 = 173 $. Also, $ (z + w)(\overlinez + \overlinew) = S + z \overlinew + \overlinez w