Nonlinear Equations & Systems

Unit 2 · Advanced Math

What the SAT asks here

Nonlinear equations include quadratic, radical, exponential, and rational equations. The SAT focuses on quadratics (factor or quadratic formula), radicals (square both sides then check for extraneous roots), and linear-quadratic systems (substitute and solve). Expect 3–4 problems per test.

Quadratic formula

Solve ax² + bx + c = 0

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant: number of real solutions

Δ = b² − 4ac > 0 → two distinct real solutions

Δ = 0 → one repeated real solution

Δ < 0 → no real solutions

SAT Tip: Try factoring first; if no clean factorization jumps out within ~15 seconds, switch to the quadratic formula.

Caution: Squaring both sides of a radical equation can introduce extraneous roots. Always substitute every candidate back into the ORIGINAL equation.

Type 1

Solve a quadratic equation

Choose factoring when the quadratic factors cleanly; otherwise apply the quadratic formula. Use the discriminant when the question asks about the number of solutions.

Example 1

Solve x² − 7x + 12 = 0.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 2

Radical equation

Isolate the radical, square both sides, solve the resulting polynomial, then substitute every candidate into the ORIGINAL equation to discard extraneous solutions.

Example 2

Solve √(2x + 3) = x for real x.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 3

Linear-quadratic system

Substitute the linear expression into the quadratic, solve the resulting quadratic, then back-substitute for y.

Example 3

Solve the system y = x² − 4 and y = 3x.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

← Equivalent Expressions Nonlinear Functions →