Step 1 :Given the cubic equation: \(x^3 + 7x^2 + (6-a)x - a = 0\)
Step 2 :Calculate the discriminant: \(Δ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2\) with \(a = 1, b = 7, c = (6 - a), d = -a\)
Step 3 :Find the value of 'a' for which the discriminant is equal to zero
Step 4 :Solve the equation: \(-27a^4 - 126a^2(6 - a) - 4a(6 - a)^3 + 1372a + 49(6 - a)^2 = 0\)
Step 5 :Find the real solutions for 'a': \(a \approx 2.383\) and \(a \approx -2.010\)
Step 6 :Calculate the sum of the real solutions: \(\boxed{0.37}\)