The equation #x^2+5x+5=0# is an example of a quadratic equation that cannot be solved by factorizing. We could solve this equation by completing the square, but this method can be quite time-consuming, especially as equations become more complex.
It is for this reason that the quadratic formula, which is derived from the method of completing the square, is commonly used to solve quadratic equations which cannot be solved by factorizing.
Quadratic formula and discriminant
The solutions of a quadratic equation in the form \[\blue ax^2+\green bx+\purple c=0\] are: