Sum of Solutions SAT Desmos: Find Sum and Product of Roots Fast

By the Cheetah Prep team · Reviewed July 13, 2026

You can use Desmos to get the sum of solutions and the product of roots fast: graph the equation, find the $x$ values where it equals $0$ , then add and multiply those $x$ values. This works well on the digital SAT because it cuts out extra algebra and lowers the chance you drop a sign or factor wrong.

Here is the quick calculator method:

Type your equation as $y =$ something, then set it equal to zero. For example, graph $y = f(x)$ and use the $x$ intercepts where $f(x) = 0$ .
Tap the intercept points (or read them from the graph) to get the solutions $x_1$ and $x_2$ (or more if the graph crosses more times).
Compute what the question asks: sum is $x_1 + x_2$ and product is $x_1x_2$ .

Two SAT tips so you do not get tricked:

Use solutions to the right equation. If the question is written like $f(x) = 7$ , graph both $y = f(x)$ and $y = 7$ , then use the intersection $x$ values, not the zeros of $f$ .
If the curve just touches the axis, that root still counts, but it is repeated. Desmos will show one touching point, so check the prompt for wording like double root.

If you want a fast refresher on reading intercepts cleanly in Desmos, use the steps in Desmos x intercepts SAT: Find Zeros Fast by Graphing.

When to use this Desmos method

Use this Desmos method when the SAT asks for the sum of solutions or the product of roots, and it does not require you to show the algebra that produces those numbers.

Use it when you can rewrite the prompt as solve for $x$ and Desmos shows the solution $x$ values clearly. Read the $x$ values, then add or multiply them.

Use it for these common question patterns:

Quadratics and other polynomials where the prompt says, "What is the sum of the solutions?" or "What is the product of the roots?" for an equation such as $f(x) = 0$ .
Equations set equal to a constant, like $f(x) = 7$ , when the question asks for the sum or product of the $x$ values that make the equation true. Graph both sides and use the intersection $x$ values.
Nonlinear equations in one variable where factoring is annoying, for example when coefficients are messy or the expression is not factor friendly.
Systems in two variables when you only care about the $x$ coordinates of intersection points, and the prompt asks you to combine those $x$ values.

Do not lean on this method when:

The prompt expects an exact expression and the graph gives a decimal that could be a rounded version.
There are more solutions than you can see in the viewing window because you forgot to zoom out or adjust the window.

If you need a wider Desmos toolkit for SAT graph moves, use the SAT Desmos guides.

Step by step in Desmos

Rewrite the prompt as an equation you can solve for x
Start by deciding what x values count as solutions. If the prompt already has something like f(x) = 0, you are looking for x intercepts. If it says f(x) = 7, you are looking for intersection x values where the two sides match.
f(x)=0
Graph both sides as two y equations
In Desmos, type each side as its own y equation. This avoids algebra mistakes and makes it clear you are solving the right equation.
y=f(x) y=7
Zoom so you can see every intersection
Drag and zoom until you can see all the places the graphs meet in the visible window. If the SAT is asking for sum or product of solutions, missing one intersection usually breaks the whole problem.
y=x^3-4x y=0
Tap each intersection to capture the x values
Click or tap each intersection point so Desmos shows its coordinates. Write down the x coordinates only, since the question is about solutions in x. If a graph just touches and turns around, that x value is still a solution even if it does not cross.
(x_1,0),(x_2,0)
Store the x values as variables so you can compute fast
After you read the x values, define them as variables on new lines. This keeps your arithmetic clean and lets you check sum and product in one place.
x_1= x_2=
Compute the sum of solutions and the product of roots
Now type the expressions you need. For sum, add the x values. For product, multiply them. Use parentheses if any x value is negative so you do not drop a sign.
S=x_1+x_2 P=x_1x_2
If there are more than 2 solutions, add and multiply them all
Some SAT problems use polynomials that cross the axis 3 or 4 times. The sum of solutions means add every distinct solution x value you found. The product of roots means multiply them all. If a root is repeated, the prompt usually tells you, but Desmos may only show one touching point, so read the wording carefully.
S=x_1+x_2+x_3 P=x_1x_2x_3

Exact expressions to enter

$y=f(x)$ Type this into Desmos
$y=0$ Type this into Desmos
Use when the equation is already set to zero, and you want x intercepts.
$y=f(x)-g(x)$ Type this into Desmos
Use when the prompt is f(x)=g(x). The solutions are the x intercepts of this difference.
$y=f(x)$ Type this into Desmos
If the prompt is f(x)=c, also enter y=c and use intersection x values.
$y=c$ Type this into Desmos
Horizontal line for equations like f(x)=c.
$x_1=\text{first x value of an intercept or intersection}$ Type this into Desmos
After you tap a point, use its x coordinate for x_1.
$x_2=\text{second x value of an intercept or intersection}$ Type this into Desmos
Repeat for x_2, and for x_3 if there are 3 solutions.
$x_1+x_2$ Type this into Desmos
Sum of solutions for 2 solutions.
$x_1x_2$ Type this into Desmos
Product of roots for 2 solutions.
$x_1+x_2+x_3$ Type this into Desmos
Sum of solutions for 3 solutions.
$x_1x_2x_3$ Type this into Desmos
Product of roots for 3 solutions.

Worked SAT style example

Example

Worked SAT style example: The equation $x^2 - 5x - 14 = 0$ has solutions $x_1$ and $x_2$ . What is $x_1 + x_2$ and what is $x_1x_2$ ?

Open Desmos and type $y = x^2 - 5x - 14$ .
Find the $x$ intercepts of the graph (the points where $y = 0$ ). Tap each intercept to read the $x$ value.
Record the solutions. You should see $x_1 = -2$ and $x_2 = 7$ .
Compute the sum: $x_1 + x_2 = -2 + 7 = 5$ .
Compute the product: $x_1x_2 = (-2)(7) = -14$ .
Quick reason this works: solutions to the equation are exactly the $x$ values where the graph crosses the $x$ axis, because that is where $y = 0$ . Once you have the solution values, sum and product are just arithmetic.

Answer:

x_1 + x_2 = 5

and

x_1x_2 = -14

Common mistakes

Most errors come from pulling the wrong $x$ values off the graph, or from combining the right values the wrong way.

Finding zeros when the equation is not $= 0$ . If the prompt is $f(x) = 7$ , the solutions are where $f(x)$ equals $7$ , not where it equals $0$ . Graph $y = f(x)$ and $y = 7$ , then use the intersection $x$ values.
Using $y$ coordinates instead of $x$ coordinates. The sum of solutions and product of roots use the solution values $x_1, x_2, \ldots$ , not the intercept height. When you tap a point, use the first coordinate.
Missing a solution because of the viewing window. If part of the graph is off screen, you can miss an intercept or an intersection. Zoom out, pan, or set a window that shows every crossing you need.
Not noticing a touch point root. If the curve touches the axis and turns around, there is still a root there. Some questions treat it as a repeated root, even though Desmos shows one point.
Rounding too early. If Desmos gives decimals, keep extra digits until the final step. Rounding $x_1$ and $x_2$ first can change $x_1 + x_2$ or $x_1x_2$ .
Multiplying when the question asks for a sum (or vice versa). Circle the exact words, then write the target expression, like $x_1 + x_2$ or $x_1x_2$ , before you compute.
Typing the expression wrong. Missing parentheses is a classic trap, for example typing $y = x + 2^2$ when you meant $y = (x + 2)^2$ . If your intercepts look weird, recheck grouping. For clean intercept reading, see SAT Desmos guides.

When this method does not work

This method does not work well when Desmos cannot show every solution clearly or when the question is testing algebra structure, not the numeric answers.

Desmos only helps if you can see and click the right points. If the graph hides a solution, your sum of solutions or product of roots will be wrong even if your arithmetic is perfect.

Watch out for these common failure cases:

Solutions are not real. If an equation has complex roots, the graph has no $x$ intercepts, so there is nothing to add or multiply.
The viewing window hides roots. A root at $x = 50$ will not appear if you are zoomed in near the origin. Zoom out and scan left and right before you commit.
Intersections are too close together. When two roots are close, the curve can look like it crosses once. Zoom in until you see separate crossings or a clear touch.
The graph touches and turns. A repeated root is easy to miss because the curve does not cross the axis.
The prompt wants an exact expression. If the answer choices are in radicals or fractions and Desmos gives a decimal, you might not be able to match it without algebra.
Domain restrictions matter. If the problem only allows certain $x$ values, the graph might show extra solutions you must ignore.

If reading intercepts cleanly is the issue, review SAT Desmos guides and practice changing the window and zoom level on purpose.

Practice questions

1.Use Desmos to solve. The equation $x^2 - 7x + 10 = 0$ has solutions $x_1$ and $x_2$ . What is $x_1 + x_2$ ?

2.Use Desmos to solve. The equation $x^2 + x - 12 = 0$ has roots $r$ and $s$ . What is $rs$ ?

3.Use Desmos to solve. Let $f(x) = x^2 - 2x - 8$ . If $f(x) = 4$ has solutions $a$ and $b$ , what is $a + b$ ?

4.Use Desmos to solve. The equation $x^3 - 4x = 0$ has three real solutions. What is the product of all real solutions?

5.Use Desmos to solve. The equation $x^2 - 6x + 9 = 0$ has solutions $x_1$ and $x_2$ (counting multiplicity). What is $x_1 + x_2$ ?

6.Use Desmos to solve. The system $y = x^2 - 5x + 4$ and $y = 0$ intersects at two points with $x$ coordinates $p$ and $q$ . What is $pq$ ?

FAQ

What does sum of solutions mean on the SAT?

It means: add the $x$ values that make the equation true. If the solutions are $x_1$ and $x_2$ , the sum is $x_1 + x_2$ . On a graph, solutions are the $x$ coordinates where the graph crosses $y = 0$ for $f(x) = 0$ . For an equation like $f(x) = 7$ , solutions are the $x$ coordinates where the graphs intersect.

Is product of roots the same as product of solutions?

Yes. A root is a solution to $f(x) = 0$ . If the solutions are $x_1$ and $x_2$ , their product is $x_1x_2$ .

How do I do sum of solutions SAT Desmos quickly?

Graph the equation so Desmos can solve for $x$ . Read the solution $x$ values from the $x$ intercepts or intersection points. Then type the sum into a Desmos expression line, for example $x_1 + x_2$ . If the equation is $f(x) = 0$ , use the $x$ intercepts. If it is $f(x) = 7$ , graph $y = f(x)$ and $y = 7$ , then use the intersection points.

What if the equation has more than 2 solutions?

Include every real solution Desmos shows as an intercept or intersection. The sum is $x_1 + x_2 + x_3 + \dots$ . The product is $x_1x_2x_3\dots$ . If the prompt says sum of all solutions, keep going past 2, even if the equation looks quadratic.

What if Desmos shows only 1 intercept, but the prompt says there are 2 roots?

That usually means the graph hits the axis and bounces back, so the root repeats. Desmos shows one point of contact, but the equation can still have a double root. If the question asks for sum or product of solutions counting multiplicity, count that root twice, for example $x_1 = x_2$ .

How do I avoid using the wrong solutions when the equation is not equal to 0?

Do not rewrite an equation into $f(x) = 0$ unless you do it correctly. For $f(x) = 7$ , solutions happen where the graph of $y = f(x)$ hits the line $y = 7$ . For $f(x) = g(x)$ , graph $y = f(x)$ and $y = g(x)$ , then take the $x$ values where they intersect.

Why does this method help on nonlinear equations in one variable?

Because you can skip algebra that eats time, like messy factoring. Desmos shows the actual solution $x$ values on the graph, so the sum and product become simple arithmetic.

What if the intercept values look like decimals?

Zoom in, then tap the point so Desmos shows the coordinates more precisely. If the answer choices are fractions or radicals, zoom in more until the value is clear. If it still shows a decimal, use that label to pick the closest exact answer choice.

How can I compute the sum and product inside Desmos after I find the solutions?

Read the solutions from the graph. Put each one on its own line as a variable, for example $a = -2$ and $b = 3$ . Then type $a + b$ to get the sum, or $ab$ to get the product. This cuts down on arithmetic mistakes, especially with negatives.

Does this work for systems in two variables too?

Yes. If the question asks for the $x$ coordinates of intersection points, graph both equations. Tap each intersection and record the $x$ values. Then add or multiply them exactly as the prompt says.

About this page: written and reviewed by the Cheetah Prep team. Last reviewed July 13, 2026.