the sum of equations of the form ${ ax }^{ 2 }+bx+2=0$ that can be formed if the equations have real roots $\left( a\underset { - }{ > } 2,b\underset { - }{ < } 6 \right) and a and b arepositive integers)$ is/are

For real roots, the discriminant of ax^2 + bx + 2 = 0 must be non-negative, meaning b^2 - 8a >= 0. Given the conditions 2 < a < 6 and b < 6 for positive integers, the possible values for a are 3, 4, and 5. Testing a = 3 requires b^2 >= 24, which is impossible for b < 6 since the maximum b = 5 gives 25 >= 24, so b = 5 works. Testing a = 4 requires b^2 >= 32, which again only allows b = 5. Testing a = 5 requires b^2 >= 40, which has no solution for b < 6. Consequently, only two equations can be formed: 3x^2 + 5x + 2 = 0 and 4x^2 + 5x + 2 = 0.

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