There are times in the life of a mathematician when he must solve a quadratic equation without the use of the quadratic formula. Typically, this occurs in math classes, but it also proves useful when dealing with polynomials of higher degree. Example 1:
(x-3)(x+1) = 0.
So we know that multiplying those two terms together produces zero. This can only happen if at least one of the terms is itself equal to zero. Therefore, either
x - 3 = 0 or
x + 1 = 0.
Anyway, you will not need to be a genius to observe that x is therefore 3 or -1. In this case, of course, it is not possible for both terms to be 0; x cannot equal 3 and -1 at the same time.
That was a rather simple example however. Example 2: What if we were to solve 6x²- 3x - 3 = 0 by factorizing. The methods for factorizing are discussed elsewhere, but for now you can take it on trust that
6x²- 3x - 3 = (2x+1)(3x-3)
If (2x+1)(3x-3) = 0, we can deduce that either
2x + 1 = 0 or
3x - 3 = 0.
Again, very basic manipulation of these equations will indicate that x = -½ or x = 1. As before, these solutions are mutually exclusive.
Example 3: For a final equation, we shall observe the equation 4x² + 4x + 1 = 0. It is no coincidence that 4x² + 4x + 1 = (2x + 1)(2x + 1) = (2x + 1)². In this case, and in others where the contents of the parantheses are identical, there is only one solution: in this case, at:
2x + 1 = 0
ie x = -½
To provide an algebraic summary, we will look at the factorization (ax + b)(cx + d). In this case:
ax + b = 0 or
cx + d = 0
In other words, x = -b/a or x = -d/c. You won't need to remember those exact equations, they are easily deriveable if necessary.
As a final note, if you interpret a quadratic equation as a graph, the roots are indeed the places where the graph cuts the x-axis, because the x-axis is the line y=0. This should help you to see why quadratic equations in general have one, two or sometimes no solutions.
Extension: If you factorize a cubic, quartic, quintic etc. into multiple brackets and are looking for roots, exactly the same method applies: that is, at least one of terms must be equal to zero. This is handy for equations of sixth degree or higher, which have no general solution.
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