Nonlinear Equations in One Variable: SAT Practice Questions & Study Guide

Nonlinear equations in one variable include quadratic equations (ax^2 + bx + c = 0), radical equations (sqrt(2x + 1) = x - 1), and rational equations (1/x + 2 = 3/x). Unlike linear equations, these can have two solutions, one solution, or no real solutions—and they require different solving strategies depending on their form. The Digital SAT tests all three types, with quadratic equations appearing most frequently.

For quadratic equations, the three main solving approaches are factoring (fastest when it works), completing the square (reliable but slower), and the quadratic formula (always works but prone to arithmetic error). The choice depends on the equation: if the coefficient of x^2 is 1 and the constant and linear terms suggest integer factors, try factoring first. If the equation is in vertex form or near it, completing the square is natural. If coefficients are large or irrational, the quadratic formula is the safest bet.

Extraneous solutions are introduced when you raise both sides of an equation to a power or multiply both sides by a variable expression. For radical equations, always square both sides to eliminate the radical, then check each solution in the original equation. If a candidate solution makes the original equation undefined or yields a different value than claimed, it is extraneous and must be discarded. The SAT tests this by including extraneous solutions among the answer choices—students who don't check lose points here.

Rational equations on the SAT typically require multiplying through by the LCD to clear denominators, converting to a polynomial equation. Again, solutions that make any original denominator zero are extraneous. Identifying the domain restrictions at the start (noting which values of x make denominators zero) helps you flag potential extraneous solutions before they fool you.