oxed{f(x) = kx^2 ext{ for some constant } k}

February 22, 2026

$oxed{f(x) = kx^2 ext{ for some constant } k}$

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a(x + y)^2 + b(x + y) + c + a(x - y)^2 + b(x - y) + c = 2(ax^2 + bx + c) + 2(ay^2 + by + c).
2a x^2 + 2a y^2 + 2b x + 2c = 2a x^2 + 2a y^2 + 2b x + 2b y + 4c.
Equating coefficients: $ 2c = 2b y + 4c $ for all $ y $, which implies $ b = 0 $ and $ c = 0 $. Thus, $ f(x) = ax^2 $. Verification confirms this satisfies the equation. Hence, all solutions are quadratic functions of the form $ f(x) = kx^2 $.
Question: Let $ a $ and $ b $ be complex numbers such that $ |a| = |b| = 1 $ and $ a
eq b $. Find the value of $ rac{a + b}{a - b} + rac{a - b}{a + b} $.
Solution: Let $ S = rac{a + b}{a - b} + rac{a - b}{a + b} $. Combine the fractions:

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