Quadratic Equations
Roots in interval (0,2); sign of expression
Grade Class 12
Question:
If both roots of the quadratic $ax^2 + bx + c = 0$ lie in $(0, 2)$ then $25ac + 20bc + 16c^2$ is always
positive
negative
equal to zero
cannot say with given information
Step-by-Step Solution
Key Concept: Evaluate $f(0)=c>0$, $f(2)=4a+2b+c>0$, and use that $25ac+20bc+16c^2=c(25a+20b+16c)$. The sign of $25a+20b+16c$ is not determined solely by the root condition.
Cannot determine sign without knowing sign of $a$ (or $c$). Answer: cannot say.
Correct Answer: 4