If both a and b belong to the set $\displaystyle \left{ 1,2,3,4 \right}$ then the number of equations of the form $ ax^{2}+bx+1=0$ having real roots is :

For the quadratic equation to have real roots, its discriminant must be non-negative, so we require b^2 - 4a >= 0. Testing the values from the set {1, 2, 3, 4} for both a and b, we find the valid pairs where b^2 is at least 4a. If b is 1, no values of a work since 1 is less than 4 times any number in the set. If b is 2, a can be 1; if b is 3, a can be 1 or 2; if b is 4, a can be 1, 2, 3, or 4. Summing these valid combinations gives 0 + 1 + 2 + 4 = 7 equations.

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