Equations with Variable in Denominator

Consider equation of the form `(p(x))/(q(x))=0`.

Solving of this equation is based on the following fact: fraction equals zero if and only if numerator equals zero and denominator doesn't equal zero (we can't divide by 0).

Therefore, we solve such equations in two steps:

  1. Solve equation `p(x)=0`.
  2. For each root of equation p(x)=0 check whether it is root of equation q(x)=0. If it is root then this root is not of initial equation, otherwise, it is root of initial equation.

Therefore equation p(x)=0 is consequence of equation `(p(x))/(q(x))=0` (after multiplying both sides by q(x)). Thus, there can appear extraneous roots. We can eliminate the with the help of condition `q(x)!=0` or by checking (direct substitution of root into initial equation).

Example. Solve equation `((3x-6)(x-5))/(x^2-x-2)=0`.

From equation `(3x-6)(x-5)=0` we find that x=2 and x=5, but when x=2 denominator `x^2-x-2` equals zero, thus x=2 is not root of the equation. When x=5, denominator `x^2-x-2` doesn't equal zero, therefore, x=5 is root of the initial equation. Therefore, there is only one root: x=5.