Advanced Math / Nonlinear Equations One Variable
Nonlinear Equations in One Variable
Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms. Useful strategies include factoring, applying the zero product property, using inverse operations, checking for extraneous solutions, and interpreting parameters from the structure of an equation.
Least you need to know
- A quadratic equation can often be solved by factoring, completing the square, or using structure from its roots.
- Squaring both sides of an equation with radicals can introduce extraneous solutions, so proposed solutions should be checked.
- For rational equations, values that make a denominator equal to zero are not allowed even if they appear after algebraic manipulation.
Key notation
- $(x-a)(x-b)=0$ — An equation whose solutions are $x=a$ and $x=b$
- $x^2+bx+c=0$ — A quadratic equation in standard form
Worked example
- Consider $\sqrt{x+10}=x-2$.
- Square both sides: $x+10=(x-2)^2=x^2-4x+4$.
- Rewrite as $x^2-5x-6=0$, so $(x-6)(x+1)=0$.
- The possible solutions are $6$ and $-1$.
- Checking in the original equation, only $x=6$ works.
Common mistakes
- Do not divide by an expression that could be zero without considering the solution that would make it zero.
- When a question asks for one solution, the equation may have more than one.
- Parameter questions often become simpler when the equation is compared to a factored form.
How to recognize it
- The problem asks for a solution, a greater or lesser solution, or the number of solutions.
- The equation may include a squared expression, a square root, a rational expression, or a parameter.
- The fastest path often uses structure rather than expanding everything immediately.
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