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Home » Alternate approach: total number of trios with exactly two close pairs. - High Altitude Science

Alternate approach: total number of trios with exactly two close pairs. - High Altitude Science

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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Find all complex numbers \( z \) such that \( f(z) = 0 \).
We solve the quadratic equation in complex numbers:
\[ z^2 + (1 + i)z + i = 0 \]
\[ z = \frac{-(1 + i) \pm \sqrt{(1 + i)^2 - 4(1)(i)}}{2} \]
First, compute the discriminant:
\[ (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i \]
\[ 4 \cdot i = 4i \]
\[ \text{Discriminant} = 2i - 4i = -2i \]
Now, compute \( \sqrt{-2i} \). Let \( \sqrt{-2i} = a + bi \), then:
\[ (a + bi)^2 = a^2 - b^2 + 2abi = -2i \]

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