Math · Advanced Math

Nonlinear Equations in One Variable and Systems of Equations on the Digital SAT

This skill covers solving quadratic and other nonlinear equations, plus systems where a line meets a parabola or another curve. You will factor, use the quadratic formula, complete the square, and reason about how many solutions an equation has using the discriminant. Systems questions usually reduce to a single quadratic after substitution.

Advanced Math is where the top of the score range gets decided, and this skill sits right in the middle of it. Students who can solve a clean quadratic but freeze when a constant is unknown, or when the question asks how many times two graphs intersect, leave real points behind here. The SAT loves reframing the same quadratic in ways that test whether you understand solutions, not just the solving steps.

A useful mindset shift: solutions of an equation are intersection points of graphs, and the discriminant b² - 4ac counts them. Positive means two real solutions, zero means exactly one, negative means none. A big share of the hard questions in this skill are just that fact wearing a costume.

New to the Digital SAT? Start with how the test is structured and scored.

What the SAT actually tests

Solving quadratic equations by factoring, taking square roots, completing the square, and the quadratic formula
Using the discriminant b² - 4ac to determine whether an equation has two, one, or zero real solutions, often with an unknown constant
Solving linear-quadratic systems by substitution, where setting the two expressions for y equal produces one quadratic in x
Interpreting solutions of a system as the intersection points of the two graphs, including tangent lines that touch a parabola exactly once
Equations with radicals or rational terms in one variable, where squaring or clearing fractions can introduce an extraneous solution you must check
Sum and product shortcuts: for x² + bx + c = 0, the solutions add to -b and multiply to c

Strategies that work

Set everything equal to zero first

Before factoring or using the formula, move every term to one side so the equation reads (quadratic) = 0. Factoring an equation like x² - 5x = -6 without moving the -6 over is a guaranteed wrong answer. The zero on one side is what makes the zero product property work.

Count solutions with the discriminant

Any question about "how many solutions" or "exactly one solution" is a discriminant question, even when it looks like a graphing question. Compute b² - 4ac, set it greater than, equal to, or less than zero depending on what is asked, and solve for the unknown constant. You almost never need the actual solutions.

Substitute, then treat it as one quadratic

For a system with y = (line) and y = (parabola), set the right sides equal and collect everything on one side. Each x you find must go back into one of the original equations to get its y. The number of x solutions is the number of intersection points.

Check answers in the original equation

When you square both sides or multiply by a variable expression, verify each candidate solution in the equation you started with. Squaring can create extraneous solutions, and the SAT writes distractors around exactly those fake roots. A ten-second plug-in protects the point you just earned.

Mistakes to avoid

Dividing both sides by x to simplify something like x² = 5x, which silently deletes the solution x = 0. Factor out x instead: x(x - 5) = 0.
Reading roots with the wrong sign from factored form: (x - 2)(x - 3) = 0 gives x = 2 and x = 3, not x = -2 and x = -3.
Sign and bookkeeping errors completing the square, like adding (b/2)² inside the parentheses without subtracting the matching amount outside.
Dropping the 4 in b² - 4ac, or forgetting that a negative c makes -4ac positive, which flips the conclusion about how many solutions exist.
Solving a system for x and stopping there when the question asks for y or for a sum of coordinates. Always reread what the question actually wants.

Solve it with Desmos

Let the calculator do the work

The Digital SAT's built-in Desmos calculator can crack Nonlinear Equations in One Variable and Systems of Equations questions in a few clicks. Here are the moves that apply, step by step.

All 15 Desmos techniques

Practice with Nova, your Math coach

Answer right here and Nova walks you through it, just like in a drill. Difficulty labels are relative within this skill.

Tip: the Calculator button on each question opens the same Desmos graphing calculator you get on the real Digital SAT. Try solving these with it.

Question 1Easy

The equation x² - 5x + 6 = 0 is given.

What are the solutions to the equation?

Pick an answer to get instant coaching from Nova.

Question 2Medium

In the equation x² + 8x + c = 0, c is a constant, and the equation has exactly one real solution.

What is the value of c?

Pick an answer to get instant coaching from Nova.

Question 3Hard

The system of equations y = x² - 2x - 3 and y = x + 1 has two solutions, (x₁, y₁) and (x₂, y₂).

What is the value of y₁ + y₂?

Pick an answer to get instant coaching from Nova.

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The rest of Advanced Math

Nonlinear Equations in One Variable and Systems of Equations is one of 3 skills in Advanced Math. The ideas build on each other, so it pays to learn them as a set rather than in isolation. See how Advanced Math fits together.

Other Math skills to explore

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