Factoring quadratics to find the roots is a skill students are expected to have as they move through mathematics. Where the zero’s of a function are, or where the x-values are when y = 0, helps determine how fast the function grows. In real-world terms, thinking about projectile motion, the roots help determine where a ball hits the ground when hit by a bat, or thrown, or hit by a golf club. Students, especially in Algebra courses, spend a lot of time working with quadratics, factoring them, determining the vertex, finding the roots, and using the quadratic formula. It is almost a given that on a standardized test students will be asked to find the roots of a given quadratic or be asked to write the factored form of the given quadratic equation, so an understanding of how the factored form, roots and graph of a quadratic equation are related is important.
As a teacher, obviously we want students to learn how to do this with and without technology. Technology helps students see how the equation, both standard and factored form, and the graph are connected and where the roots are and how the factored form relates to the roots. I know as a teacher, I spent time helping students learn to factor and find roots both by hand and graphically. Using a calculator to find the roots of a quadratic helped students explore quadratics, allowing them to answer more in depth questions because they could get to the roots quickly with the calculator, see the graphical representation, and make a relationship with the factored form of the equation. The calculator was a great tool for helping students make conjectures about roots of a given quadratic and then quickly test their predictions. It gave them confidence in their factoring and understanding of quadratics.
Knowing that there are both TI and Casio users out there, I have made a quick video that demonstrates how to find the roots of a quadratic with both the Casio Prizm and the TI-84+ CE. Hopefully this will be helpful to you.