Why do you get 2 solutions in the quadratic equation?

Answer 1

This is due to the zero-product property. In principle, any polynomial equation of degree 2 can be factored as:

(x - a)(x - b) = 0

Here is a specific example:

(x - 5)(x + 3) = 0

Now, if the product of two factors is zero, it follows that one of the two factors is equal to zero; so the above becomes:

x - 5 = 0 OR x + 3 = 0

Solving the individual parts, you get the two solutions. Of course, it is possible that the two factors happen to be the same; in this case, the polynomial is said to have a "double" root (i.e., a double solution).

Similarly, a polynomial equation of degree 3 can be separated into 3 factors, a polynomial of degree 4 can be factored into 4 factors, etc.