Destruction Games That Will Melt Your Brain—Explosions, Crashes, and Chaos!
Destruction Games That Will Melt Your Brain—Explosions, Crashes, and Chaos!
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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 + 
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