For what values does the equation: X2 – 2(a-1)x + (4a-3)=0 have real roots? For what integer values of a are the roots real integers?
Use the discriminant. It must be > or = to zero for the roots of the equation to be real. You'll get a quadratic equation for the discriminant. Wherever this equation is zero or positive (the y value), the roots of the original equation will be real.