Theory of Equations 1 Question 42
43. If $P(x)=a x^{2}+b x+c$ and $Q(x)=-a x^{2}+b x+c$, where $a c \neq 0$, then $P(x) Q(x)$ has atleast two real roots.
$(1985,1 M)$
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Solution:
- $P(x) \cdot Q(x)=\left(a x^{2}+b x+c\right)\left(-a x^{2}+b x+c\right)$
Now, $\quad D _1=b^{2}-4 a c$ and $D _2=b^{2}+4 a c$
Clearly, $\quad D _1+D _2=2 b^{2} \geq 0$
$\therefore$ Atleast one of $D _1$ and $D _2$ is (+ ve). Hence, atleast two real roots.
Hence, statement is true.
