Advanced Math / Nonlinear Equations One Variable
Nonlinear Equations in One Variable
Nonlinear equations in one variable include quadratic, radical, rational, absolute value, exponential, and polynomial equations. Solving often requires factoring, taking square roots, isolating a radical, using exponent rules, or checking for restrictions and extraneous solutions.
Least you need to know
- Quadratic equations may have two, one, or no real solutions.
- Factoring a product equal to zero lets you set each factor equal to zero.
- Radical and rational equations may require checking domain restrictions or extraneous solutions.
Key notation
- $x^2+bx+c=0$ — A quadratic equation in one variable
- $\sqrt{x}$ — The principal square root of $x$
Worked example
- Consider $x^2-9x+20=0$.
- Factor: $(x-4)(x-5)=0$.
- Set each factor equal to zero: $x=4$ or $x=5$.
- The solutions are 4 and 5.
Common mistakes
- A quadratic may have two solutions; read whether the question asks for a specific one, a sum, a product, or any solution.
- Squaring both sides of an equation can introduce extraneous solutions.
- Values that make a denominator zero are not allowed.
How to recognize it
- The equation contains powers, roots, absolute values, variables in denominators, or variables in exponents.
- The problem may ask for one solution, the greater or lesser solution, a sum or product of solutions, or a value that satisfies the equation.
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