# Nonlinear Equations and Systems on the SAT: Examples & Practice Source: https://1600.now/sat-skill/nonlinear-equations-and-systems --- Math · Advanced Math ## Nonlinear Equations and Systems Choose a method that fits the equation: factor a quadratic, isolate a radical before squaring, split an absolute value into cases, or substitute to find graph intersections. Check candidates in the original problem. Written by [Luke Finigan](https://1600.now/about) 2 min read Updated Oct 2, 2026 ### Factor a quadratic and use the zero-product rule For $x^2-5x+6=0$, factor to $(x-2)(x-3)=0$. A product is zero when at least one factor is zero, so the solutions are 2 and 3. If asked for the sum, answer 5; if asked for the product, answer 6. Move all terms to one side before using this rule. From $(x-2)(x-3)=4$, you cannot set the individual factors equal to zero. ### Squaring creates candidates that need a check For $\sqrt{x+6}=x$, the right side must be nonnegative. Squaring gives $x+6=x^2$, or $(x-3)(x+2)=0$. The candidates are 3 and $-2$, but only 3 satisfies the original equation. At $x=-2$, the left side is 2 and the right side is $-2$. A graph or a squared equation can suggest a candidate; substitution into the unsquared equation settles whether it works. ### Keep domain restrictions in rational and absolute-value equations For $\frac{x+1}{x-2}=3$, exclude $x=2$ before multiplying. Solving gives $x+1=3x-6$, so $x=\frac{7}{2}$, which is allowed. A value making an original denominator zero must always be rejected. For $|2x-1|=5$, solve $2x-1=5$ and $2x-1=-5$, giving 3 and $-2$. An absolute value cannot equal a negative number, so $|2x-1|=-5$ has no solution. ### Substitute to solve a mixed system For $y=x^2$ and $y=x+2$, set $x^2=x+2$. Factoring gives $(x-2)(x+1)=0$. The intersections are $(2,4)$ and $(-1,1)$ after substituting into either equation. A system can have more than one intersection. If graphing, inspect the relevant domain and both branches rather than stopping at the first visible point. ### Check your understanding Try it yourself: Which value solves $\sqrt{x+2}=x$? · A $-1$ B $0$ C $2$ D $-1$ and $2$ ### Sample nonlinear equations and systems questions Nonlinear Equations and Systems · Easy $6x(x-4)(x+8)=0$ What is one solution to the given equation? A -4 B 0 C 6 D 8 - [Open the question and explanation](https://1600.now/bank/math/748bc300) Nonlinear Equations and Systems · Easy: What is the $y$-intercept of $y=8x^2+5x+4$ in the $xy$-plane? · A $(0, 4)$ B $(0, 5)$ C $(0, 8)$ D $(0, 9)$ - [Open the question and explanation](https://1600.now/bank/math/c6b27c20) Nonlinear Equations and Systems · Easy $\sqrt{w} + 18 = 30$ What is the solution to the given equation? A $4$ B $6$ C $24$ D $144$ - [Open the question and explanation](https://1600.now/bank/math/6b582db1) ### Practice this skill 1. Classify the equation before choosing a method. 2. Write restrictions before clearing denominators or squaring. 3. Check every candidate and answer the exact requested root, coordinate, sum, or product. [Practice nonlinear equations and systems](https://1600.now/bank/math/skill/Nonlinear%20equations%20in%20one%20variable%20and%20systems%20of%20equations%20in%20two%20variables) [Print the nonlinear equations and systems worksheet and worked answers](https://1600.now/sat-nonlinear-equations-and-systems-worksheet). Then use a [mixed practice module](https://1600.now/modules) to check whether you can choose this method among other question types. ### Source and question labels The bank label for this guide is "Nonlinear equations in one variable and systems of equations in two variables" in Advanced Math. The worked examples above are written for this guide. - [College Board: Types of Math tested](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types) ### Related skills - **[Nonlinear Functions](https://1600.now/sat-skill/nonlinear-functions)** The form of a nonlinear function tells you what you can read directly. - **[Equivalent Expressions](https://1600.now/sat-skill/equivalent-expressions)** Equivalent expressions give the same value for every input where both are defined. - **[Systems of Linear Equations](https://1600.now/sat-skill/systems-of-linear-equations)** A solution to a system must satisfy both equations.