Nonlinear Equations — SAT Math Explained
Equations that, when graphed, do not produce a straight line. They include quadratic, exponential, radical, rational, and absolute value equations — any equation where the variable is raised to a power other than 1 or appears in a non-linear way.
The Core Idea
Nonlinear equations can have multiple solutions, no solutions, or solutions that require careful checking because algebraic manipulation can introduce extraneous solutions.
Types
ax² + bx + c = 0 — solved by factoring, completing the square, or quadratic formula
√x = 5 — isolate the radical, square both sides; ALWAYS check for extraneous solutions
Equations with variables in denominators — multiply by LCD; check for undefined values
|x - 3| = 5 — splits into two cases: x - 3 = 5 or x - 3 = -5
2ˣ = 16 — rewrite with same base or use logarithms
Extraneous Solutions
False solutions introduced by algebraic manipulation (especially squaring both sides). Always substitute all solutions back into the ORIGINAL equation to check.
General Strategy
1. Identify the type of nonlinear equation
2. Isolate the nonlinear term (radical, absolute value, etc.)
3. Undo the nonlinear operation (square both sides, split absolute value, etc.)
4. Solve the resulting equation
5. Check ALL solutions in the original equation — reject extraneous ones
Common Errors to Avoid
Forgetting to check for extraneous solutions after squaring
Only finding one solution for absolute value equations (there are usually two)
Not checking for undefined values in rational equations (denominators equal to zero)
Practice: Nonlinear Equations
5 SAT-style questions. Select your answer and get an instant explanation.
Which of the following is NOT a nonlinear equation?
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