First slide

Theory of equations

Question

Let $f (x) = {ax}^{2} + bx + c, a, b, c \in R .$ If f (x) takes real values for real values of x and non-real values for non-real values of x, then

Moderate

A

a = 0
B

b = 0
C

c = 0
D

nothing can be said about a, b, c.

Solution

Suppose $a \neq 0$ . We rewrite f(x) as follows:
$\begin{array}{l} f (x) = a \{x^{2} + \frac{b}{a} x + \frac{c}{a}\} \\ = a \{{(x + \frac{b}{2 a})}^{2} + \frac{4 ac - b^{2}}{4 a^{2}}\} \end{array}$
$\begin{aligned} f (- \frac{b}{2 a} + i) = a \{{(- \frac{b}{2 a} + i + \frac{b}{2 a})}^{2} + \frac{4 ac - b^{2}}{4 a^{2}}\} \\ = a \{- 1 + \frac{4 ac - b^{2}}{4 a^{2}}\}, which is a real number \end{aligned}$
This is against the hypothesis. Therefore, a = 0.

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