What counts as a nonlinear equation on the SAT?

Any equation where the variable is squared, cubed, under a root, or sitting in an exponent. Quadratics dominate, with exponential equations and mixed line and parabola systems close behind.

Should I factor or use the quadratic formula?

Try factoring first, since it is faster when it works. If nothing factors cleanly within about ten seconds, switch to the formula or to Desmos rather than forcing it.

How does the discriminant help me?

The value of b² − 4ac tells you how many real solutions exist before you solve anything. Positive means two, zero means exactly one, negative means none, and the SAT asks this directly.

How do I solve an exponential equation without logarithms?

Rewrite both sides over a matching base and set the exponents equal. The SAT designs these so a common base always exists, which keeps logarithms off the table.

Do the practice questions here get harder?

You choose the level. Each generator produces easy, medium, and hard versions, so you can climb from clean factoring to gnarly discriminant work at your own pace.

Advanced Math · Skill 2 of 3

SAT Nonlinear Equations Practice

When x picks up a square or climbs into an exponent, you are in nonlinear territory. The digital SAT asks you to solve quadratics, match exponential forms, and untangle a line crossing a parabola. See the recurring shapes below, follow one full solution, then drill versions that never repeat.

Domain: Advanced Math
About 35% of the test is Advanced Math
Difficulty: easy to hard
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Drill nonlinear equations now → Browse Advanced Math

The question types that keep returning

Once you can name the setup, half the work is done. These four cover the territory.

Pattern 01

Quadratics that factor

Set everything equal to zero, factor, and apply the zero product rule. On the SAT the factor pairs are usually small integers, which is a strong hint to try factoring before anything heavier.

Pattern 02

Quadratic formula and the discriminant

When factoring stalls, the formula finishes the job. The discriminant b² − 4ac also answers a favorite SAT question all by itself: the sign of that single number counts the real solutions for you.

Pattern 03

Exponential equations

Solve 3^{x + 1} = 27 by rewriting 27 as 3³, then equating exponents: x + 1 = 3, so x = 2. Matching the bases converts an exponential equation into a linear one.

Pattern 04

A line meets a parabola

Substitute the linear expression into the quadratic and solve what remains. Two roots mean two crossing points, and a discriminant of zero means the line just grazes the curve.

A complete solution, no steps skipped

Worked example · medium

A ball is launched from a rooftop, and its height in feet after t seconds is modeled by h(t) = −16t² + 48t + 64. After how many seconds does the ball hit the ground?

Hitting the ground means the height is zero: −16t² + 48t + 64 = 0.
Divide every term by −16 to shrink the numbers: t² − 3t − 4 = 0.
Factor: (t − 4)(t + 1) = 0, so t = 4 or t = −1. A flight cannot land before it launches, so discard t = −1.
Confirm t = 4 in the original: −16 × 16 + 48 × 4 + 64 = −256 + 192 + 64 = 0. The model agrees.

Answer: t = 4 seconds

Rockets, dropped phones, diving dolphins: the projectile changes, the factoring never does.

The traps built into these questions

Keeping impossible roots. Factoring gives t = 4 and t = −1, but time cannot run backward. Context questions expect you to throw out the root the story forbids.
Solving for the wrong height. Hitting the ground means h = 0. Returning to launch height means h equals the starting value. A sloppy read of the target wastes a clean solve.
Dividing away a solution. Dividing x² = 5x by x quietly deletes x = 0. Move everything to one side and factor instead: x(x − 5) = 0 keeps both roots.
Discriminant sign slips. In b² − 4ac, squaring a negative b still yields a positive number. One dropped sign turns two solutions into none.

Desmos as your second solver

Move everything to one side, graph it, and the x intercepts are your solutions, no quadratic formula required. For a line crossing a parabola, type both and count the intersection points on screen. On the hard end of module two, this is often the difference between finishing and guessing.

Why drilling here is different

One quadratic is a puzzle; two hundred fresh ones are training. These nonlinear equation generators mint new discriminants, bases, and intersection setups on every run, at easy, medium, and hard. Underneath sits Satified's bank of 1,483 questions, every answer independently verified before you ever see it.

Crack the curve before test day.

Drill nonlinear equations →

What test takers ask about this skill

What counts as a nonlinear equation on the SAT?: Any equation where the variable is squared, cubed, under a root, or sitting in an exponent. Quadratics dominate, with exponential equations and mixed line and parabola systems close behind.
Should I factor or use the quadratic formula?: Try factoring first, since it is faster when it works. If nothing factors cleanly within about ten seconds, switch to the formula or to Desmos rather than forcing it.
How does the discriminant help me?: The value of b² − 4ac tells you how many real solutions exist before you solve anything. Positive means two, zero means exactly one, negative means none, and the SAT asks this directly.
How do I solve an exponential equation without logarithms?: Rewrite both sides over a matching base and set the exponents equal. The SAT designs these so a common base always exists, which keeps logarithms off the table.
Do the practice questions here get harder?: You choose the level. Each generator produces easy, medium, and hard versions, so you can climb from clean factoring to gnarly discriminant work at your own pace.

Keep the momentum

Solving curves pairs naturally with reading them. These two round out the Advanced Math domain.