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/ Now complete the square for $ z $:
Now complete the square for $ z $:
February 22, 2026
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4x^2 + 4y^2 + 4z^2 = 16 - 8z + z^2 \Rightarrow 4x^2 + 4y^2 + 3z^2 + 8z - 16 = 0.
Divide by 4 to simplify:
x^2 + y^2 + rac{3}{4}z^2 + 2z - 4 = 0.
rac{3}{4}z^2 + 2z = rac{3}{4}(z^2 + rac{8}{3}z) = rac{3}{4}\left( \left(z + rac{4}{3}
ight)^2 - rac{16}{9}
ight) = rac{3}{4}\left(z + rac{4}{3}
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