How can you determine the number of solutions for an equation?

Answer 1

In general, you cannot: it all depends on the domain.

x + 2 = 0 has no solutions is the set of positive integers but does have one if the domain is the integers.

2x - 3 = 0 has no solutions if the domain is integers, but there is one solution if the domain is the rationals.

x2 - 2 = 0 has no solutions if the domain is the rationals but there are two solutions if the domain is the reals.

x2 + 2 = 0 has no solutions if the domain is the reals but there are two solutions if the domain is the complex numbers.

Cos(x) = 1 has no solutions if the domain is (0, 360) but two solutions for the domain [0, 360].

Answer 2

The number of solutions for an equation can be determined by analyzing the degree of the equation and its graphical representation. For a linear equation, there is either one solution (if the lines intersect) or no solution (if the lines are parallel). Quadratic equations can have two solutions, one solution, or no real solution, depending on the discriminant. Higher degree equations can have multiple solutions or no solutions depending on the nature of the equation.