No, not all quadratic equations may be solved by factoring.
For example,
[tex] x^{2} -2x+2=0[/tex]
This may not be factored. In fact, the solutions to this equation are imaginary numbers. Factoring this requires a complex technique, so the easiest way to solve for the roots is to use the quadratic formula.
Understanding the nature of roots of quadratic equations proves this. Only those with perfect square discriminants may be factored.
Given ax² +bx + c = 0
[tex]discriminant = b^{2} -4ac[/tex]
*The discriminant is the part inside the radical sign of the quadratic formula.
For the given example above,
[tex]discriminant = -2^{2} -4(1)(2)=4-8=-4[/tex]
The discriminant here is negative, thus it is not a perfect square.
