For a quadratic equation ax^2 + bx + c = 0, first find the signs of the roots using the following rules:

1. If a and c have opposite signs, the roots have opposite signs.

2. If a and c have the same sign, the roots have the same sign.

Also, when this is the case, if a and b have opposite signs, the roots are positive, else they are negative.

Next, using the Diagonal Sum Method of solving quadratic equations the denominator of the roots is taken as the factor set of a and the numerator of the roots is taken as the factor set of b.

Using this a probable list of roots is created. To choose the correct roots from this, the sum of the product of the numerator and the denominator of the roots should be equal to -b.

For example take the quadratic equation: x^2 + 4x + 4 = 0. As a and c have the same sign, the roots have th same sign and as a and b have the same sign they are negative.

Now the factors of a are 1 and the factors of c are 1, 2 and 4

Using these we can create the probable root sets: (-4/1, -1/1) and (-2/1 and -2/1)

Taking the diagonal sum: -5, -4

As -4 = -b, the roots are (-2, -2)