quadratic

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Joined: Aug 4, 2011

by hemangsarkar, Sep 13, 2012, 5:16 AM

Q) prove that no such quadratic $f(x) = ax^2 + bx + c$ (where $a,b,c$ are real) can exist which takes real values for real values of $x$ and imaginary values for imaginary values of $x$ .

solution : let us assume that such a quadratic exists.

$f(x) = ax^2 + bx + c = a \left(\left (x+\frac{b}{2a} \right)^2 + \frac{4ac - b^2}{4a^2} \right)$

but $f \left(i - \frac{b}{2a} \right)$ is a real number. $(i^2 = -1)$
hence no such quadratic exists.

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