Rational Equations

Equation `f(x)=g(x)` is called rational, if both f(x) and g(x) are rational expressions. If f(x) and g(x) are integer expressions, then equation is called integer; if at least one of expressions f(x), g(x) is fractional, then rational equation `f(x)=g(x)` is called fractional.

For example, linear and quadratic equations are integer.

Example. Solve equation `2/(2-x)+1/2=4/(x(2-x))`.

Domain of this equation is all x, except x=2 and x=0.

Common denominator is `2x(2-x)`.

Therefore equation can be rewritten as `(2*2x)/((2-x)*2x)+(1*x(2-x))/(2*x(2-x))=(4*2)/(x(2-x)*2)`.

This can be rewritten as `(4x+x(2-x))/(2x(2-x))=8/(2x(2-x))`, or `(4x+2x-x^2-8)/(2x(2-x))=0`.

Therefore, `4x+2x-x^2-8=0` or `x^2-6x+8=0`. Roots of this equation are x=2 and x=4, but x=2 is not in the domain, thus, there is only one root of the initial equation: x=4.