First slide

Theory of expressions

Question

If $f (x) = a x^{2} + b x + c, g (x) = - a x^{2} + b x + c$ where $a c \neq 0$ then $f (x) g (x) = 0$ has

Moderate

A

at least three real roots
B

no real roots
C

at least two real roots
D

two real roots and two imaginary roots

Solution

Let $D_{1}$ and $D_{2}$ be discriminates of $a x^{2} + b x + c = 0$
and $- a x^{2} + b x + c = 0$ respectively. Then,
$D_{1} = b^{2} - 4 a c, D_{2} = b^{2} + 4 a c$
Now, $a c \neq 0 \Rightarrow$ either $a c > 0$ or $a c < 0$
If $a c > 0,$ then $D_{2} > 0$ Therefore, roots of $- a x^{2} + b x + c = 0$ are real .
If $a c > 0,$ then $D_{1} > 0$ Therefore, roots of $a x^{2} + b x + c = 0$ are real .
Thus , $f (x) g (x)$ has at least two real roots.

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