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How to Solve SAT Quadratic Equations with Desmos

Last updated: October 5, 2026

Type the quadratic into Desmos and click the points where the parabola crosses the x-axis. Each crossing is a solution, and Desmos labels it with exact coordinates. No factoring, no quadratic formula, and the number of crossings tells you how many solutions there are.

The problem

Solve: x² − x − 6 = 0
Solution: x = −2 and x = 3

What do you type?

One row: x^2 − x − 6

Desmos turns the ^2 into x² and draws a parabola opening upward. You don't need to type y = in front; an expression in x on its own gets graphed as y. You don't need the = 0 either. "Equals zero" means "where is y zero," and y is zero on the x-axis.

Click the parabola. Gray dots appear at the points Desmos marks for you. Click the two on the x-axis and they read (−2, 0) and (3, 0). The x-coordinates are the answer: x = −2 and x = 3.

Check one in the row below if you want: type (−2)^2 − (−2) − 6 and Desmos prints 0.

What if the equation isn't set to zero?

The SAT often prints it as x² = x + 6. That's the same equation, but you don't have to move everything to one side first. Graph each side as its own row:

y = x²
y = x + 6

Click where the parabola and the line cross. Desmos labels (−2, 4) and (3, 9). Again the x-coordinates are the solutions, −2 and 3. The y-values are only the height where the two sides meet, and they're not part of the answer.

This is the graph-both-sides move from solving linear equations in Desmos. A curve on one side doesn't change it.

How many solutions are there?

Count the crossings:

• Two crossings: two real solutions
• Touches the axis once: one solution
• Never reaches the axis: no real solutions

Change the −6 to +6 and the parabola lifts off the x-axis entirely. No gray dots on the axis, no real solutions. You can see the answer to a "how many solutions" question before you work out any algebra.

When the question hides a constant, like "for what value of c does x² − x + c = 0 have exactly one solution," type x^2 − x + c. Desmos offers to add a slider for c. Drag it until the bottom of the parabola just touches the x-axis and read the value of c off the slider. If the slider's steps skip past the exact value, click the slider's number and type a candidate from the answer choices instead.

Why the count works out that way, and the discriminant that predicts it without a graph, is the strategy side of this skill. SAT Nonlinear Equations and Systems covers it.

What if the answer has a square root in it?

Desmos labels crossings in decimals. Graph x² − 2x − 1 and the crossings read 2.414 and −0.414, but the answer choices say 1 ± √2.

Don't convert by hand. Type each answer choice on its own row, 1 + sqrt(2), and Desmos prints its decimal value next to it. The choice that matches the crossing is the answer.

What about the sum or product of the solutions?

Click both crossings, then do the arithmetic on what Desmos shows you. For x² − x − 6 = 0, the sum is −2 + 3 = 1 and the product is −2 × 3 = −6. When a question asks for "the positive solution" or "the greater solution," it's asking for one of the two dots. Read both, then pick.

How do you practice this?

Take 10 quadratic questions from practice tests: some set to zero, some not, a couple that ask how many solutions. For each one:

1. Type it in, as one expression or as two sides, before you try any algebra
2. Click every crossing and write the x-coordinates down, not the y-coordinates
3. Count the crossings and check that the count matches what the question asked for

When you get one wrong, check which you did: read a y-value instead of an x-value, or miss a crossing that was off the screen. Zoom out once before you commit.

HIROSCORE tracks your accuracy on nonlinear equations as its own skill, separately from the rest of Advanced Math, so a habit of reading the wrong coordinate shows up instead of hiding inside a math score that looks fine. The GPS for your SAT score.