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Let \( a \) and \( b \) be complex numbers such that \( a + b = 4 \) and \( a^2 + b^2 = 10 \). Find \( |a - b| \).
\( a + b = 4 \)
\( a^2 + b^2 = 10 \)
We use the identity:
Substitute:
4^2 = 10 + 2ab \Rightarrow 16 = 10 + 2ab \Rightarrow 2ab = 6 \Rightarrow ab = 3
Now compute \( |a - b| \). Note:
|a - b|^2 = (a - b)(\overline{a} - \overline{b}) = |a|^2 - a\overline{b} - \overline{a}b + |b|^2
But better to use algebra:
|a - b|^2 = (a - b)^2 \quad ext{if real, but not necessarily.}