# How to Solve Rational Equations

Rational Equation Solver
X +
——————————
X +
= X +
——————————
X +

Rational functions are fractions with polynomials in the numerators and denominators. In algebra, it is easy to solve rational equations when the numerator and denominator of a rational function are first-degree polynomials, i.e. a linear functions. A typical rational equation has the form

(ax+b)/(cx+d) = (ex+f)/(gx+h)

where x is the variable and the parameters a through h are constants. By cross multiplying the numerators and denominators, you can turn the original equation into a quadratic equation. The outline below will show you the algebraic steps to solve these kinds of math problems, or you can use the rational equation solver calculator on the left.

#### Step 1

Cross-multiply both sides of the equation (ax+b)/(cx+d) = (ex+f)/(gx+h), thereby transforming it into the equivalent equation

(ax+b)(gx+h) = (ex+f)(cx+d).

#### Step 2

Use the distributive property to expand both sides of the equation into

agx2 + (bg+ah)x + gh = cex2 + (cf+ed)x + fd.

Then combine like terms to form the quadratic equation

(ag-ce)x2 +(bg+ah-cf-ed)x + gh-fd = 0.

#### Step 3

If the coefficient ag-ce is not zero, use the quadratic formula to solve for x. You will obtain either two real values (possibly repeated roots) or two complex numbers. If the coefficient ag-ce is zero, you will have a linear equation, with possibly zero, one, or infinitely many solutions.

Plug each solution back into the original equation (ax+b)/(cx+d) = (ex+f)/(gx+h). If a solution yields a denominator of zero, that solution must be discarded. Some rational equations may have no solutions.