First slide

Theory of expressions

Question

If $α, β \in R$ are the roots of the $a x^{2} + b x + c = 0, k \in R$ lies between $α$ and $β,$ if

Moderate

A

$a k^{2} + b k + c < 0$
B

$a^{2} k^{2} + a b k + a c < 0$
C

$a^{2} k^{2} + a b k + a c > 0$
D

none of these

Solution

Let $f (x) = a x^{2} + b x + c$ It is given that $α$ and $β,$ roots of $f (x) = 0$ So, k lies between $α$ and $β,$ if
$a f (k) < 0 \Rightarrow a (a k^{2} + b k + c) < 0 \Rightarrow a^{2} k^{2} + a b k + a c$

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