Quadratic Inequalities

Algebra Level 5

Consider the following system of inequalities:

\[\begin{cases} (c-1)x^2+2cx+c+4\leq 0\\ cx^2+2(c+1)x+(c+1)\geq 0 \end{cases}\]

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

Details and assumptions

\(c\) can be negative.

The system has a unique solution if there is only 1 real value \(x\) which is satisfied throughout.


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