There is little doubt that for many digital SAT math problems, graphing is the most expeditious approach when using Desmos for assistance. Each situation is a bit different, but there can be some drawbacks to using graphing for solving certain types of equations, and in some cases, you may find that you prefer regression. When seeking both solutions to a quadratic equation, graphing is generally the better choice, as you can typically enter the equation unaltered, then you can view the relevant coordinates of points on the resulting graph to get the answers you need. However, there are some types f problems involving solving quadratic equation that require you to match answers in fraction form, which coordinate values don't use, or in a compound form such as -3 + √11, necessitating the evaluation of each answer choice to find a match with the decimal value displayed for coordinates. It's also possible you'll need to do some arithmetic with one or both of the solution values, so you'd have to read them off the graph and type them in; if any such values have decimals that repeat or that have more digits than Desmos will display (Desmos will display three decimals by default, and four if you zoom in far enough), you could end up with an imprecise result. Regression doesn't suffer from any of these shortcomings, so though solving a quadratic equation for both solutions using regression is an advanced technique, you might want to master it.

As explained previously, a regression statement will find only one value for each regression parameter (variable with unknown value). In the large majority of cases, that is sufficient for enabling you to answer a digital SAT math question using a direct conversion of a given (or modeled) equation into regression form, even when there might be two solutions, as is often the case with quadratic equations and other types of equations as well.

The problem might have a restriction that results in only a single solution, and you can include that restriction in your regression statement; the problem might ask for any solution, so whichever one Desmos gives you will qualify; the problem could indicate that there is only a single solution, using wording such as "what is the solution," which tips you off.

However, in a small minority of situations, you will need to find two solutions to a quadratic equation in order to be certain you can answer the question. This can be straightforward, as when a problem asks for both solutions, and it can also happen when the problem gives a list of possible solutions and it asks you to select the one that is valid. Even though in the latter case you are only being asked for one solution, you have to know what both of them are; if you attempt a simple regression that just represents the quadratic equation as given, you'll only get one of the solutions, and it's a toss-up as to whether it will be the solution that's among the answer choices.

Here's an example where using straightforward regression will find one of the solutions, but it's the wrong one; the solution Desmos produces isn't in the list of answer choices:

Using regression to find the values of both solutions to a standard form quadratic equation

In order for a regression to produce two solutions for one unknown in an equation, each of the two solutions must be represented by a distinct regression parameter (variable). Therefore, we must craft an equation, or a system of equations, that includes both solutions. The technique we have created for this situation when it involves a quadratic equation is to represent the equation as a system of two equations: one for the sum of the solutions, and one for the product of the solutions.

You should know that in a standard form quadratic equation ax² + bx + c = 0, the formula -b / a gives the sum of the solutions, and the formula c / a gives the product of the solutions. Therefore, if we label the solutions as p and q, we can form this system of equations:

We can represent this system in a Desmos regression statement the same way we would with any system of equations: place the expressions on the left side of the equations into a list on the left side of the regression statement, and place the expressions on the right side of the equations into a list on the right side of the regression statement. Desmos will find the values of the regression parameters p and q that make both equations valid, and, thus, we will have both solutions to the quadratic equation:

An important note about crafting the regression: because both expressions on the right side of the equations in the system have a denominator of a, we can pull that out and make it the denominator of a fraction that has the list of numerators as its numerator; we don't need to repeat " / a" for both expressions in the list on the right-hand side. Here's the optimized form, then, of a regression statement that will find both solutions for a quadratic equation in standard form:

This clever technique lets you use Desmos, which does not do algebra, as a quadratic equation solver without resorting to the far more tedious, time-consuming, and error-prone technique of entering the quadratic formula (twice).

Here's the previous example solved using this sum/product method; this time, we see both solutions, so we can select the answer choice that represents a solution:

Solving standard form quadratic equations with answer choices that include radicals

There are occasionally digital SAT math problems that require you to solve a quadratic equation, but the answer asks for an expression of the form h ± √r or some variation, where h represents the x-coordinate of the vertex or axis of symmetry of the parabola corresponding to the quadratic expression and √r represents the distance between the axis of symmetry and the roots:

These are the forms of solutions that would be produced by solving with completing the square; the quadratic formula will also produce solutions that can be manipulated to match these forms. But if you want to use Desmos, there is an advanced regression technique we've devised that you could use if you want to become a true Desmos ninja. You can set up a system of equations using the sum/product technique described above, but you'll have to use a different representation of the two solutions. If one solution is h + √r and the other solution is h - √r, the sum of the solutions would be h + √r + h - √r = 2h, and the product of the solutions would be (h + √r)(h - √r) = h² - r, so the system of equations would be

The corresponding regression statement template would be

This will solve for h and r, allowing you to construct the two solutions h ± √r: