Parameter range for specific root conditions (location/count)
Determine parameter values so that roots satisfy location constraints (e.g., both roots ≥ 4, roots in an interval, one root between given bounds) or a specified number of distinct real roots.
For how many values of $a$ is the equation $$(x - a)(x^2 - x + a) = 0$$ satisfied by exactly two distinct values of $x$? A $0$ B $1$ C $2$ D $3$ E $4$ F more than 4
The equation $x ^ { 4 } + b x ^ { 2 } + c = 0$ has four distinct real roots if and only if which of the following conditions is satisfied? A $b ^ { 2 } > 4 c$ B $b ^ { 2 } < 4 c$ C $c > 0$ and $b > 2 \sqrt { c }$ D $c > 0$ and $b < - 2 \sqrt { c }$ E $\quad c < 0$ and $b < 0$ F $\quad c < 0$ and $b > 0$