I am a supporter of the Common Core State Standards for Mathematics (CCSSM). While far from perfect, I think the CCSSM Mathematical Practices are downright beautiful. I see shades of these practices in my classroom from time to time. I am striving to develop these behaviors, like constructing viable arguments and critiquing the reasoning of others, in all of my students.
While reading through the discussion on Algebra (page 62 of the CCSSM document) earlier today, I found the following:
An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form...
...The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b_1+b_2)/2)h, can be solved for h using the same deductive process.
Rewriting an expression can often provide greater understanding of its underlying structure. As a student, I had a rough experience with geometry in high school; however, I did very well in algebra class. Dynamic software provides a learner like me the opportunity to see the connections between algebra and geometry. I can bootstrap my way up to better geometric understanding by seeing the connections between symbolic manipulation and graphs.
Below is an image of a Geogebra sheet I made which demonstrates how we can rewrite the quadratic formula as the axis of symmetry of the parabola plus or minus a distance.
This sheet demonstrates the graphical connection between the quadratic formula and the roots. Symbolically, we can rewrite the quadratic formula as two fractions with denominator 2a.
The first fraction, the green -b/(2a), gives the axis of symmetry of the parabola. The second orange fraction gives the distance we walk to the left and right of the axis of symmetry. We can think of an x-intercept in the following way: start at the axis of symmetry, then walk the same distance (the second fraction) in the positive or negative direction along x.
If this discriminant b^2-4ac is positive (the radical stuff), there are two distinct real roots.
If this discriminant is zero, there is one repeated real root.
If this discriminant is negative, then there are two imaginary roots. Graphically, there are no real solutions, demonstrated by the Geogebra sheet.
This approach demonstrates visually the utility of the discriminant in determining the number of roots of a parabola written in standard form.