Treat an algebraic fraction exactly like a numeric fraction, but factorise first. Simplify by cancelling common factors (never individual terms), add or subtract over a common denominator, multiply and divide by factorising then cancelling across the fractions, and solve equations by multiplying every term by the common denominator to clear the fractions before solving normally.
The single biggest mistake is cancelling terms instead of factors. Students see (x + 3) / (x + 5) and try to "cancel the x", which is meaningless: you can only cancel a factor that multiplies the whole numerator and the whole denominator, never a term that is added or subtracted inside a bracket.
The fix is a habit, not a trick: factorise the numerator and denominator fully before you do anything else, on every single algebraic fraction question. Once everything is in factorised form, the fraction rules you already know from Primary school (common denominators to add, flip and multiply to divide) work exactly the same way.
Step 1
Factorise every numerator and denominator fully
Step 2
Cancel factors common to numerator and denominator
Step 3
Add/subtract: find the LCD, rewrite each fraction over it
Step 4
Multiply/divide: factorise, flip if dividing, cancel, then multiply
Simplify (x² + 5x + 6) / (x² − 4).
Solution
x² + 5x + 6 = (x + 2)(x + 3) (two numbers that multiply to 6 and add to 5 are 2 and 3).x² − 4 is a difference of two squares, so x² − 4 = (x + 2)(x − 2).(x + 2)(x + 3) / (x + 2)(x − 2).(x + 2), since it appears in both the numerator and the denominator: (x + 3) / (x − 2), valid for x ≠ 2, x ≠ −2.The step students get wrong
Cancelling x from (x + 3) / (x − 2) to get 3 / −2 is not allowed. x is a term inside each bracket, not a factor of the whole numerator or the whole denominator. Only a complete factor, such as (x + 2) above, can be cancelled, and it must appear as a factor of the entire top and the entire bottom.
Express as a single fraction: 3 / (x + 1) + 2 / (x − 1).
Solution
(x + 1) and (x − 1) share no common factor, so the lowest common denominator is found the same way you'd find the HCF and LCM by prime factorisation: their product: (x + 1)(x − 1).3(x − 1) / (x + 1)(x − 1) + 2(x + 1) / (x + 1)(x − 1).[3(x − 1) + 2(x + 1)] / (x + 1)(x − 1).3x − 3 + 2x + 2 = 5x − 1.(5x − 1) / (x + 1)(x − 1), which can also be written as (5x − 1) / (x² − 1).Quick check: substitute a simple value, such as x = 2, into both the original expression and your final answer. Original: 3/3 + 2/1 = 1 + 2 = 3. Final answer: (10 − 1)/(4 − 1) = 9/3 = 3. They match, so the simplification is correct.
Express as a single fraction: 4 / (x + 3) − 1 / (x + 3)².
Solution
(x + 3) and (x + 3)². Since (x + 3)² already contains (x + 3) as a factor, the LCD is simply (x + 3)², not the product of both.(x + 3): 4(x + 3) / (x + 3)². The second fraction is already over the LCD.[4(x + 3) − 1] / (x + 3)² = (4x + 12 − 1) / (x + 3)² = (4x + 11) / (x + 3)².Multiplication and division follow the same numeric rules you already know: multiply straight across, or flip the second fraction and multiply for division. The only new step is factorising everything first, so that common factors can be cancelled before you multiply out.
Example: multiplying (x² − 9) / (x + 4) × (x + 4) / (x + 3)
x² − 9 = (x + 3)(x − 3), a difference of two squares.(x + 3)(x − 3) / (x + 4) × (x + 4) / (x + 3).(x + 4) across the two fractions, and cancel (x + 3) across the two fractions: only (x − 3) is left.(x − 3), valid for x ≠ −4, x ≠ −3.Example: dividing (x + 2) / 3x ÷ (x + 2)² / 6x²
(x + 2) / 3x × 6x² / (x + 2)².(x + 2) on top against the (x + 2)² on the bottom, which leaves a single (x + 2) in the denominator. Then simplify the numbers and powers of x: 6x² / 3x = 2x.2x / (x + 2), valid for x ≠ 0, x ≠ −2.When the question says "solve" instead of "simplify" or "express as a single fraction", the goal changes: clear every fraction by multiplying both sides of the equation by the common denominator, then solve the resulting equation normally, using the same steps as solving linear equations. The same clearing-the-fraction move also appears when you need to change the subject of a formula that contains a fraction.
Example: solving 2/(x − 1) + 1/x = 1
x(x − 1). Multiply every term by it.2x + (x − 1) = x(x − 1).2x + x − 1 = 3x − 1. Expand the right side: x² − x. The equation becomes 3x − 1 = x² − x.x² − 4x + 1 = 0. This does not factorise over integers, so use the quadratic formula to get x = 2 + √3 or x = 2 − √3, which is x = 3.73 or x = 0.268 to 3 significant figures.The step students get wrong
Forgetting to multiply the non-fraction terms by the common denominator too. In the example above, the 1 on the right must also be multiplied by x(x − 1), becoming x(x − 1), not left as 1. Every single term in the equation gets multiplied, not only the fractions.
State the restrictions: whenever a variable appears in a denominator, that value cannot be a solution, since division by zero is undefined. State the restricted values (for example x ≠ 0) as part of a full answer, and always check your final solutions against them.
Can I cancel across an addition or subtraction sign?
No. Cancelling only works on factors, meaning terms joined by multiplication, not on terms joined by addition or subtraction inside a bracket. If your expression has a plus or minus sign that is not inside a factorised bracket, you cannot cancel across it until it has been factorised.
How do I know if I need a common denominator or need to factorise first?
Do both, in that order. Always factorise every denominator first, since a factorised form often reveals that two denominators already share a common factor, which keeps the lowest common denominator as small as possible instead of multiplying every denominator together unnecessarily.
What is different about equations with fractions compared to simplifying fractions?
Simplifying keeps the expression as a fraction throughout. Solving an equation lets you clear the fractions entirely by multiplying every term by the common denominator, because you are working with an equation (both sides equal), not a single expression, so multiplying both sides by the same non-zero quantity keeps it balanced.
Why does the quadratic formula appear when solving fraction equations?
Clearing the fractions in an equation with variables in more than one denominator often produces a quadratic equation. Solve it the same way as any other quadratic: try factorising first, and fall back to the quadratic formula or completing the square when it does not factorise over integers.
— Mr Gan Math Tuition
Mr. Gan works with students who want a factorise-first habit that removes the guesswork from algebraic fractions for good.
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