Why doesn't an equation cross the x-axis when it cannot be factored?

Answer 1

Since I can't show you graphs on Answers I must ask you to visualise a parabola that does cross the x-axis at two points. (It doesn't matter whether the parabola is 'open' at the top or at the bottom.) For example, we could consider the parabola

5x2 + 9x - 2

We find that it crosses the x-axis at x = 1/5 and x = -2. We call these solutions or roots of

5x2 + 9x - 2 = 0, and we can show that each of these solutions can be used to create a factor of the original parabola.

x = 1/5 yields the factor x - 1/5 :

which we demonstrate by dividing 5x2 + 9x - 2 by x - 1/5 to get 5x + 10 even (and we can check that x - 1/5 multiplied by 5x + 10 yields the original parabola).

x = -2 yields the factor x + 2 :

which we again demonstrate by dividing 5x2 + 9x - 2 by x + 2 to get 5x - 1 even.

The point of this is that when a parabola crosses the x-axis it has solutions that yield factors. However, if it doesn't cross the x-axis it cannot have solutions (because it cannot 'equal' zero), and therefore cannot be factored.