Nonlinear equations in one variable and systems of equations in two variables

Solve quadratic, absolute value, radical and rational equations, and systems that pair a line with a curve. It is part of Advanced Math in the Math section. The book teaches it in Chapter 21, Nonlinear Equations and Systems, with 19 practice questions, and tests it 10 more times across the two full-length practice tests.

About Chapter 21

A linear equation usually leads straight to one answer. A nonlinear equation can have zero, one, two or more real solutions, and some values your algebra produces may not work at all. This chapter gives you one method for every type. The College Board calls the skill "Nonlinear equations in one variable and systems of equations in two variables." It belongs to the Advanced Math domain, about 35 percent of the Math section.

What the question looks like

Most questions give an equation or a system and ask for its solutions in a specific form:

  • "What is the positive solution to the given equation?" (often a student-produced response)
  • "What are all the solutions to the given equation?"
  • "How many distinct real solutions does the given equation have?"
  • "If (x, y) is a solution to the given system, what is the value of y?"
  • "In the given equation, k is a constant. The equation has exactly one real solution. What is the value of k?"

Context questions, such as a thrown ball's height or a growing bacteria culture, ask when a quantity reaches a given value.

How to recognize it

Look at where the variable sits. If it is squared, under a root, in a denominator, between absolute value bars or in an exponent, the equation is nonlinear. A system belongs here when at least one of its equations is nonlinear, most often a line paired with a parabola.

Chapter 20 rewrites expressions without solving anything, and Chapter 22 asks about a function's features, such as its vertex or growth rate. If the task is to find the values of x that make an equation true, or the points where two graphs meet, you are in this chapter.

Common traps

  • Dividing both sides by x: You solved x^2 = 5x and got only x = 5. Factor instead: x(x − 5) = 0 gives 0 and 5.
  • Keeping only the positive root: x^2 = 49 has two solutions, 7 and −7. Write ± whenever you take a square root.
  • Keeping an extraneous solution: You squared both sides or cleared a denominator. Substitute every candidate into the original equation.
  • Factoring before one side is 0: (x − 2)(x + 1) = 4 does not mean x − 2 = 4. Expand, subtract 4, then factor.
  • Sign slip with −b: The formula begins with −b. If b = −5, the numerator begins with +5.
  • Answering the wrong quantity: The question wants the positive solution, a sum, or y. Underline the last sentence.
  • Forgetting a case: An absolute value equation with a positive right side usually gives two candidates.

Video lessons

Two short lessons from independent YouTube creators, chosen for this skill. The notes are ours; the videos belong to their creators, who are not affiliated with this site.

  • Quadratic equations (Carmel Valley Test Prep, 14:40). Watch for: Factoring, the quadratic formula and the discriminant.
  • Quadratic word problems (Penguin SAT Prep , 12:01). Watch for: What the vertex, the zeros and the y-intercept mean in context.

Practice this skill

The full method, worked examples and 19 practice questions with every answer choice explained are in Chapter 21 of the book and in the study workspace, where an AI coach can talk you through each one.

Independence

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