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Nonlinear Equations for the SAT: Complete Study Guide + Free Practice Problems
Everything the SAT tests about nonlinear equations in one place: solving quadratics by factoring, the quadratic formula, completing the square, the discriminant, radical equations, and rational equations — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupWhat are nonlinear equations?
A nonlinear equation is any equation where the variable is not simply raised to the first power — it includes squares, square roots, or the variable in a denominator. On the SAT, this almost always means quadratic equations (variable squared), along with radical equations and certain rational equations that reduce to quadratics.
| Type | Example | Typical solutions |
|---|---|---|
| Quadratic | x² − 5x + 6 = 0 | Up to 2 real solutions |
| Radical | √(x + 3) = 5 | Must check for extraneous solutions |
| Rational | 3/(x−1) = x | Cross-multiply, then solve like a quadratic |
Unlike linear equations, quadratics can have zero, one, or two real solutions. Before you commit to an answer, always check whether the question is asking for one solution, all solutions, or the sum/product of the solutions.
Solving quadratics by factoring
When a quadratic factors cleanly, factoring is the fastest way to solve it. Set the equation to zero, factor the expression, then use the zero product property: if two factors multiply to zero, at least one of them must equal zero.
Solve: x² − 2x − 15 = 0
Every quadratic must be set equal to zero before factoring. If you see x² − 2x = 15, move the 15 over first — factoring x(x − 2) = 15 directly and setting each factor equal to 15 is a common and incorrect shortcut.
The quadratic formula
When a quadratic doesn't factor nicely, the quadratic formula solves any equation in the form ax² + bx + c = 0.
Solve: 2x² + 3x − 5 = 0
The quadratic formula always works, even when factoring is hard to spot. If you spend more than 10–15 seconds hunting for factors, switch to the formula instead of guessing.
Completing the square
Completing the square rewrites a quadratic into vertex form, which is especially useful when the SAT asks for the vertex or asks you to rewrite the equation in a specific form.
Rewrite x² + 8x + 10 = 0 by completing the square.
The discriminant
The expression under the square root in the quadratic formula, b² − 4ac, is called the discriminant. It tells you how many real solutions a quadratic has without solving it fully.
| Discriminant | Number of real solutions |
|---|---|
| b² − 4ac > 0 | Two distinct real solutions |
| b² − 4ac = 0 | Exactly one real solution (a repeated root) |
| b² − 4ac < 0 | No real solutions |
When the SAT asks "for what value of k does this equation have exactly one solution," it's asking you to set the discriminant equal to zero and solve for k — no need to fully solve the quadratic.
Radical equations
To solve a radical equation, isolate the radical, then raise both sides to a power that eliminates it. This step can introduce extraneous solutions — answers that satisfy the squared equation but not the original.
Solve: √(x + 7) = x + 1
Squaring both sides of an equation can create solutions that don't actually work in the original equation. Always substitute every candidate solution back into the original radical equation before finalizing an answer.
Rational equations
Some rational equations reduce to a quadratic once you clear the denominators. Cross-multiply or multiply every term by the common denominator, then solve the resulting polynomial equation as usual.
Solve: 6/x = x − 1
Any value that makes the original denominator equal zero cannot be a solution, even if it appears after solving. Always check your candidate solutions against the original equation's restrictions.
Test day strategy for nonlinear equations
| Question signal | Fastest approach |
|---|---|
| Quadratic that looks factorable | Try factoring first — faster than the formula when it works |
| Quadratic with ugly coefficients | Go straight to the quadratic formula |
| "How many real solutions" question | Calculate the discriminant only — don't solve the full equation |
| Equation with a square root | Isolate the radical, square both sides, then check for extraneous solutions |
| Equation with a variable in the denominator | Clear denominators, solve the resulting quadratic, then check restrictions |
| "Rewrite in the form..." question | Complete the square to reach vertex form |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Nonlinear Equations 1
Factoring, the quadratic formula, and completing the square.
Start quiz → Quiz 2Nonlinear Equations 2
The discriminant, radical equations, and rational equations.
Start quiz →