Question

If $a{x}^2-bx+c=0$ have two digtinct roots lying in the interval $(a,1),a,b,cϵN$ them find the value of ${\log }_{x}abc$?

Answer 1

According to the problem:

For consistency of the problem assume $a>0$.

Therefore,

$c=0$ or $b=0$ does not have roots of $f\left(x\right)=a{x}^2-bx+c$ in $(0,1)$

Therefore $c\ge 1$ and $b\ge 1$

${\log }_x\left(abc\right)>2\iff {\log }_x\left(abc\right)>{\log }_x\left(52\right)\iff abc>25.$