Bounding A Triangle

Geometry Level 4

For \(a> 0 \), the above diagram shows the vertical line \(x=a\) bounds a triangle with the lines \(y=2x-1\) and \(y = -\frac x2 + 1 \).

If the constraint \(a>0\) is relaxed, then there is another possible value of \(a\) such that the enclosed area of the new triangle is also 5.

If the sum of these two values of \(a\) can be expressed as \( \dfrac BC\), where \(B\) and \(C\) are coprime positive integers, find \(B+C\).

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