Solve Rational Equations | Step-by-Step Calculator with Exact Simplified Solutions

A rational equation is an equation containing at least one rational expression. To solve:

Find the least common denominator (LCD) of all fractions.
Multiply both sides by the LCD to clear denominators.
Solve the resulting polynomial equation.
Exclude any solutions that make any denominator zero.

The calculator below solves equations of the form \( \dfrac{A}{x+B} + \dfrac{C}{x+D} = E \). Exact solutions are given as simplified fractions or simplified radicals.

✧ Rational Equation Solver ✧

Solve: \( \dfrac{a}{x+b} + \dfrac{c}{x+d} = e \)

Equation: \( \dfrac{a}{x+b} + \dfrac{c}{x+d} = e \)

\( a \) (numerator 1)

\( b \) (denom constant 1)

\( c \) (numerator 2)

\( d \) (denom constant 2)

\( e \) (right side)

\( \dfrac{3}{x+2} + \dfrac{-1}{x-3} = 2 \)

⚠️ Denominators cannot be zero. Solutions making \( x = -b \) or \( x = -d \) are excluded.

📐 Solution(s)

\( x = 4 \pm \sqrt{5} \)

📖 Step-by-Step Solution

STEP 1: Multiply all terms by the product of denominators \((x+b)(x+d)\)

STEP 2: Simplify and note domain restrictions (\(x \neq -b, x \neq -d\))

STEP 3: Expand and write in standard quadratic form \(Ax^2 + Bx + C = 0\)

STEP 4: Solve the quadratic equation (exact simplified values)

💡 Check Domain Restrictions