Converting a vertex form to a factored form can be achieved using a similar approach to converting from the factored form to the vertex form. To demonstrate, we are going to start with an example of an equation (Eq. 1).

Eq. 1: \(y=(x+3.5)^2-0.25\)

First of all, you need to see if there is a factored form. There is a factored form if the k value of the expression is negative, and the a value is positive, or the k value is positive and the a value is negative. As the a and k values are 1 and -0.25 respectively, it is possible to factorize this expression.

The k value (-0.25) is the y value of the vertex, and the x-intercepts (which are defined in the factored form) are where the value of the y is 0. Therefore the difference between these values is the distance that the parabola must travel in order for the vertex to meet the x-intercepts. We also have to account for the a value of the expression, which describes the stretch of the parabola. We use both of these to calculate the effective change in the y value between the x value of the vertex and the x value of the x-intercepts. Lastly, since the change in the y value is defined based on the square of the change in the x value, and we are identifying the change in x value from a given change in y, we also need to find the square root of our adjusted change in y value. This gives us an equation of (Eq. 2) to determine the difference in x value between our vertex and our x intercepts.

Eq. 2: \(\sqrt{1(0-(-0.25))}=0.5\)

Once we have the change in x required for the parabola to meet the x-axis from the vertex, we have to identify where these x-intercepts are. To do this, we have to add and subtract this change in x value to and from the x value of the vertex (Eq. 3 and Eq. 4). To get the x value of the vertex, we use the inverse of the h value of the vertex form (-3.5).

Eq. 3: \(-3.5+0.5=-3\)

Eq. 4: \(-3.5-0.5=-4\)

This leaves 2 x-intercepts which can be placed into the factored form of the equation. The x-intercepts are in the factored form of the equation as the values which added or subtracted from the x terms, where the values are the inverse of the x-intercepts (Eq. 5).

Eq. 5: \((x+3)(x+4)\)

This makes the final factored form of the quadratic expression from the vertex form.