If $a, b, c \in R$ are such that $4 a+2 b+c>0$ and $a x^2+b x+c=0$ has no real roots, then the value of…
If $a, b, c \in R$ are such that $4 a+2 b+c>0$ and $a x^2+b x+c=0$ has no real roots, then the value of $(c+a)(c+b)$ is
- greater than ab
- less than bc
- greater than ca
- less than ab + bc + ca
Solution
Since, the equation $a x^2+b x+c=0$ have no real roots and $4 a+2 b+c>0$
$
\begin{aligned}
\therefore \quad a+b+c & >0 \\
& \left\{\because a x^2+b x+c>0, \forall x \in R\right\}
\end{aligned}
$
So,
$
\begin{aligned}
\text { So, } & & c+a>-b \text { and } c+b & >-a \\
\Rightarrow & & & (c+a)(c+b)>a b
\end{aligned}
$
Asked in: AP EAMCET 2018 (22 Apr Shift 2)
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