First slide

Theory of equations

Question

If $P (x) = a x^{2} + b x + c$ and $Q (x) = - a x^{2} + d x$
$+ c, where a, b, c \in R, then P (x) Q (x) = 0 has$

Moderate

A

no real root
B

exactly two real roots
C

at least two real roots
D

none of these

Solution

Let $D_{1} = b^{2} - 4 a c$ and $D_{2} = d^{2} + 4 a c$
We have $D_{1} + D_{2} = b^{2} + d^{2} \geq 0$
$\Rightarrow$ at least one of $D_{1,} D_{2} > 0$
$\Rightarrow$ one of $P (x) = 0 or Q (x) = 0$
has real roots. Thus, $P (x) Q (x) = 0$
has at least two real roots.

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