Consider the quadratic equation (c-5)x^2-2cx+(c-4)=0, c5.

Question

Consider the quadratic equation $(c-5)x^2-2cx+(c-4)=0$, $c
e5$. Let $S$ be the set of all integral values of $c$ for which one root lies in $(0,2)$ and the other in $(2,3)$. Then the number of elements in $S$ is:

Accepted Answer

Conditions $f(0)>0$, $f(2)<0$, $f(3)>0$ give $c\in\left(\dfrac{49}4,24\right)$. Integral values are $13,14,\dots,23$, giving 11 elements.

Solutions · 1