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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.

နောက်တစ်ခုပြန်ကြည့်ရန် အကြံပြုထားသော သင်ခန်းစာ

ဤခေါင်းစဉ်ကို Nonlinear Equations in One Variable ဖြင့် ဆက်လက်သင်ယူပါ။

Nonlinear Equations in One Variable

ဆက်စပ်သင်ခန်းစာများ

တူညီသော ခေါင်းစဉ်အတွင်း အနီးစပ်ဆုံး သင်ခန်းစာများနှင့် ဆက်လက်သင်ယူပါ။

Exponential Equations in One Variable Nonlinear Equations in One Variable Nonlinear Equations in One Variable

လေ့လာရန် အခြားနည်းလမ်းများ

ဤခေါင်းစဉ်ကို လေ့လာရန် ဤခေါင်းစဉ်ခွဲကို လေ့လာရန် သင်ခန်းစာအားလုံးကို ကြည့်ရှုရန်
လေ့ကျင့်မှု စတင်ရန်