What is a rational equation?
A rational equation (also called a fractional equation) has the unknown in the denominator (bottom) of at least one fraction. Examples: 3/x = 1/(x − 4) and x/(x + 1) + 2/(x − 1) = 4/(x² − 1).
Not rational: x/3 + x/5 = 8, because the bottoms are just numbers. That is an ordinary linear equation.
Banned values (the domain)
Division by zero has no meaning. So any value of x that makes a denominator zero is not allowed. Find them first. For x/(x + 1) + 2/(x − 1) = 4/(x² − 1), the banned values are x = −1 and x = 1. The allowed values form the domain of the equation.
How to solve: Gate, Clear, Solve, Check
- Gate: factorise every denominator and write the banned values.
- Clear: multiply every term on both sides by the LCD. All the bottoms cancel.
- Solve the equation that is left. It is usually linear or quadratic.
- Check: remove any root that is a banned value. Write the answer.
Worked: x/(x + 1) + 2/(x − 1) = 4/(x² − 1). Since x² − 1 = (x − 1)(x + 1), the LCD is (x − 1)(x + 1). Multiply: x(x − 1) + 2(x + 1) = 4 → x² + x − 2 = 0 → (x + 2)(x − 1) = 0. Roots −2 and 1. But 1 is banned. Answer: x = −2.
Shortcut for one fraction on each side
If a/b = c/d, you may cross-multiply: ad = bc. Still check the banned values.
Special case: a fraction equal to zero
A/B = 0 exactly when A = 0 and B ≠ 0. Example: (x² − 9)/(x − 3) = 0 gives x = ±3, but x = 3 is banned, so x = −3.
Equivalent equations and extraneous roots
Two equations are equivalent if they have exactly the same solutions. Adding the same number to both sides, or multiplying by a non-zero number, keeps an equation equivalent.
Multiplying by an expression with x (like the LCD) is different. For some x the expression is 0, and multiplying by 0 can make a false statement look true. So the new equation may have an extra root that the original does not have. Such a root is called extraneous (outside). That is why the check step is not optional.
An equation can even have no solution: 1/(x − 1) + 1/(x + 1) = 2/(x² − 1) leads to 2x = 2, x = 1, which is banned.
Equations reducible to quadratics
Some equations become quadratic after a clever substitution.
- Biquadratic: x⁴ − 5x² + 4 = 0. Put t = x²: t² − 5t + 4 = 0, so t = 1 or t = 4. Then x² = 1 or x² = 4, giving x = ±1, ±2. (If t comes out negative, x² = t has no real root.)
- Repeated block: (x² + x)² − 8(x² + x) + 12 = 0. Put t = x² + x: t = 2 or t = 6, then solve x² + x = 2 and x² + x = 6.
- A fraction and its reciprocal: x/(x + 1) + (x + 1)/x = 5/2. Put t = x/(x + 1): t + 1/t = 5/2 → 2t² − 5t + 2 = 0 → t = 2 or t = 1/2. Then solve each simple equation for x and check the banned values 0 and −1.
Word problems with rates
Use time = distance ÷ speed and work rate = 1 ÷ time.
Boat and river
Boat speed in still water v km/h, river current 2 km/h. Upstream speed v − 2, downstream speed v + 2. If 12 km up and 12 km back takes 2.5 h: 12/(v − 2) + 12/(v + 2) = 2.5. Clear: 12(v + 2) + 12(v − 2) = 2.5(v² − 4) → 5v² − 48v − 20 = 0 → (5v + 2)(v − 10) = 0. v = 10 km/h (v = −0.4 makes no sense for a speed).
Working together
If A takes a hours and B takes b hours, together they do 1/a + 1/b of the job per hour. Time together t satisfies 1/a + 1/b = 1/t.
Always check that the answer makes sense in the story: speeds positive, times positive, numbers of people whole.
Key formulas and definitions
- Banned values: every x that makes a denominator 0
- Clear fractions: multiply every term by the LCD
- a/b = c/d ⇒ ad = bc (b, d ≠ 0)
- A/B = 0 ⇔ A = 0 and B ≠ 0
- Extraneous root: a root of the cleared equation that is a banned value
- Biquadratic ax⁴ + bx² + c = 0: put t = x²
- time = distance ÷ speed; upstream v − c, downstream v + c
- Work together: 1/a + 1/b = 1/t
Worked examples
1. Solve 6/x = 2.
Banned: x ≠ 0. Multiply by x: 6 = 2x, so x = 3. 3 is allowed. Answer x = 3.
2. Solve 3/x = 1/(x − 4).
Banned: 0 and 4. Cross-multiply: 3(x − 4) = x → 3x − 12 = x → 2x = 12 → x = 6. Allowed. Answer x = 6.
3. Solve (x + 3)/(x − 1) = 2.
Banned: 1. Multiply by (x − 1): x + 3 = 2x − 2 → x = 5. Check: 8/4 = 2 ✓. Answer x = 5.
4. Solve x²/(x − 2) = 4/(x − 2).
Banned: 2. Multiply by (x − 2): x² = 4 → x = 2 or −2. x = 2 is banned (extraneous). Answer x = −2.
5. Solve x/(x + 1) + 2/(x − 1) = 4/(x² − 1).
Banned: ±1. LCD (x − 1)(x + 1). x(x − 1) + 2(x + 1) = 4 → x² + x − 2 = 0 → (x + 2)(x − 1) = 0. x = 1 is banned. Answer x = −2.
6. Solve 1/x + 1/(x + 2) = 4/3.
Banned: 0, −2. LCD 3x(x + 2). 3(x + 2) + 3x = 4x(x + 2) → 6x + 6 = 4x² + 8x → 4x² + 2x − 6 = 0 → 2x² + x − 3 = 0 → (2x + 3)(x − 1) = 0. x = 1 or x = −3/2, both allowed.
7. Solve x⁴ − 13x² + 36 = 0.
Put t = x²: t² − 13t + 36 = 0 → (t − 4)(t − 9) = 0. x² = 4 → x = ±2; x² = 9 → x = ±3. Four roots: ±2, ±3.
8. A pipe fills a tank in 3 h, another in 6 h. How long do both take together?
1/3 + 1/6 = 1/t → 2/6 + 1/6 = 3/6 = 1/2 → t = 2 hours.
Common mistakes
- Multiplying only some terms by the LCD. Every term on both sides, including whole numbers, must be multiplied.
- Skipping the check and keeping an extraneous root that makes a denominator zero.
- Cancelling a factor with x across an equation (for example dividing both sides by x) and losing a real root.
- Using the product of all denominators when they share factors, which makes the algebra much longer; factorise first and use the lowest common denominator.