SAT Nonlinear Equations and Systems Questions
Quadratics and other nonlinear equations, and how many solutions they have.
In short
- Nonlinear Equations and Systems is part of Advanced Math, about 35% of the scored Math questions: 13 to 15 on every test.
- Math questions from all four areas are mixed in each module and arranged from easiest to hardest.
- Square-root final step is the trap Rivie's practice tests set most often in Nonlinear Equations and Systems questions.
Common traps
Most common first, from Rivie's ten practice tests.
Square-root final step
Solving x² = k yields x = ±√k. The wrong choice keeps only the positive root.
The usual mistake: You take the square root and forget the negative branch; the trap is always the +√k value.
Extraneous root
Squaring or rationalizing both sides can introduce solutions that don't satisfy the original equation; the wrong choice keeps both algebraic solutions instead of testing them back in.
The usual mistake: You solve √(x+3) = x − 1 algebraically, get two roots, and pick both, without checking which one actually satisfies the original (un-squared) equation.
Absolute-value two cases
|expression| = k splits into expression = k and expression = −k. The wrong choice solves only one case.
The usual mistake: You strip the bars, solve once, and pick that single root, missing the second branch.
Square root: missed ±
Solving x² = 25 gives x = ±5, not just x = 5. The wrong choice keeps only the positive root and misses one valid solution (or one part of an interval).
The usual mistake: You take the square root of both sides and drop the negative case because 'square roots are positive'. That's the principal root; the equation has two solutions.
Try a question
From Rivie's practice tests. Pick an answer to see why it is right or wrong.
If (x − 5)² = 49, what are the two solutions of the equation?
Video lessons
Free on YouTube, picked for this topic.