How to Solve Rational Equations

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.

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