Non Linear Equations in One Variable and System of Equations in Two Variables
Solving quadratic, absolute value, radical, and nonlinear system equations
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You get a quadratic, a radical, or a system with curves. The question asks: solve it, or tell me how many solutions exist.
Why this matters
This skill covers quadratics, absolute values, radicals, and mixed systems. There are five question types, each requiring a different technique.
The five patterns
Solve System of Equations
At least one equation is nonlinear. Substitute to reduce to one variable, then solve the resulting quadratic or simpler equation.
›Number of Solutions
Use the discriminant (b² − 4ac) to determine if a quadratic has zero, one, or two real solutions — without actually solving.
›Solve Nonlinear Equations
Solve quadratics (factoring, formula, completing the square), absolute value equations, and radical equations. Always check for extraneous solutions.
›Rearranging Formulas
Isolate a variable buried in a formula. Reverse operations step by step — the algebra is the same whether the formula is linear or not.
›Interpret Graphs
Two functions are graphed. The solution to the system is where they intersect. Read the coordinates off the graph.
The biggest trap: forgetting to check for extraneous solutions after solving radical or absolute value equations. The algebra gives you a number, but plugging it back in proves it doesn't work. The SAT counts on you skipping that step.