Quadratic Inequalities

Algebra Level 5

Consider the following system of inequalities:

{(c1)x2+2cx+c+40cx2+2(c+1)x+(c+1)0\begin{cases} (c-1)x^2+2cx+c+4\leq 0\\ cx^2+2(c+1)x+(c+1)\geq 0 \end{cases}

The sum of all real values of cc, such that the system has a unique solution, can be written as ab \frac{a}{b} , where aa and bb are coprime positive integers. What is the value of a+ba+b?

Details and assumptions

cc can be negative.

The system has a unique solution if there is only 1 real value xx which is satisfied throughout.


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