主题 7Advanced Math

Nonlinear equations in one variable

Strategies for solving equations that aren't linear — quadratics, absolute values, radicals, rationals, and higher-degree polynomials.

核心知识

Every nonlinear equation type has its own unlocking move: quadratics need factoring or the quadratic formula, absolute values split into two cases, radicals require squaring (then checking), rationals require clearing denominators (then checking domain), and higher-degree polynomials reduce through factoring. The SAT tests whether you pick the right move and whether you catch extraneous solutions.

深入理解

Nonlinear equations are not one-size-fits-all. The degree, radical sign, or absolute value bars change what you can do and what can go wrong.

On the SAT, the main equation types are:

  • Quadratic equations (degree 2): can have 0, 1, or 2 real solutions. You solve by factoring, completing the square, or the quadratic formula.
  • Absolute value equations: the absolute value creates two cases. Each case is its own linear equation.
  • Radical equations: isolate the radical, square both sides, then check your answers — squaring can introduce false solutions.
  • Rational equations: multiply through by the LCD to clear fractions, then check that your answers don't make any denominator zero.
  • Polynomial equations (degree 3+): factor out common terms or group to reduce the problem to simpler factors.

Across all these types, the SAT rewards two habits: choosing the most efficient solving method, and verifying that your answers actually work in the original equation.

知识点教程

6
1

Solve quadratic equations (factoring, completing the square, or quadratic formula).

Quadratics can be solved by factoring, the quadratic formula, or completing the square.

2

Determine the number of real solutions to a quadratic (e.g., via discriminant or graph).

The discriminant tells you how many real roots a quadratic has without solving it.

3

Solve absolute value equations.

Absolute value equations split into cases after you isolate them.

4

Solve radical equations and check for extraneous solutions.

Radical equations need squaring and a final check for extraneous solutions.

5

Solve rational equations and respect domain restrictions.

Rational equations need the LCD and a domain check before you accept any answer.

6

Solve polynomial equations using factoring where applicable.

Higher-degree polynomials usually solve by factoring and setting each factor to zero.