Factored Form | Richard Crooks's Website

In the factored form of the quadratic expression, the expression is in the form of two (or more) brackets.

$$y = a(x+b)(x+c)$$

In the factored form, the $b$ and $c$ coefficients describe where the x-intercepts of the parabola are (Figure 1).

An animated GIF of a quadratic function in the factored form showing how the x intercepts change as the b and c values change — Figure 1: The $b$ and $c$ values of the quadratic expression in the factored form reflect the x-intercept values. The x intercepts are always $-1b$ and $-1c$ in this form, whereas the location of the vertex, or the steepness of the parabola cannot be predicted by the equation in this form.

The $a$ coefficient on the other hand describes how steep the parabola is (Figure 2). However unlike the vertex form of the quadratic equation, it isn't as clear to see the relationship between this value and how steep the curve is.

An animated GIF of a quadratic function in the factored form showing how the x intercepts change as the a value changes — Figure 2: The $a$ value changes the location of the vertex, and the direction which the parabola opens in. Note that it does not define where the vertex is, as in the factored form, this is also dependant on the location of the roots (or x-intercepts) of the equation.

The factored form is useful for analyzing where a parabola crosses the x-intercepts. This can be useful if the x-intercept is significant, this could be for example, if the x-intercept intercept is the ground, and the expression describes a flight path of an object for example.

Not every expression has a factored form, some expressions do not have x-intercepts, and so they don’t have a factored form. In these cases another form of the expression is more appropriate to describe the parabola.