High School Mathematics - 2 12.8 Quadratic Equations - Discriminant
The roots of a quadratic equation whether equal, unequal or non-real depend upon the quantity b^{2}-4ac. which is called discriminant.
If b^{2}-4ac = 0, then each root of the equation becomes -b/2a and hence roots are real and equal.
If b^{2}-4ac is positive and a perfect square then, Öb^{2}-4ac is rational and hence the roots of the equation are rational and unequal.
If b^{2}-4ac is positive but not a perfect square then, Öb^{2}-4ac is real but irrational and hence the roots of the equation are irrational and unequal.
If b^{2}-4ac is negative and a perfect square then, Öb^{2}-4ac is imaginary and hence no real roots exist.
We further observe that sum of two roots in any case is -b/a and product is c/a.