Algebraic Fractions
Simplifying, adding, subtracting and solving algebraic fractions, and algebraic proof.
Learn
Algebraic Fractions, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Simplify algebraic fractions by factorising and cancelling
Factorise top and bottom, then cancel whole brackets that match: = = . Only common FACTORS cancel - never cancel individual terms across a + or −.
Multiply and divide algebraic fractions
Multiply straight across, factorising first so you can cancel early and keep things small. To divide, flip the second fraction and multiply - exactly as with numbers: ÷ = × = .
Add and subtract algebraic fractions with different denominators
Build a common denominator by multiplying the bottoms, adjusting each top to match: + = = . When subtracting, bracket the whole second numerator before expanding.
Solve equations that involve algebraic fractions
Multiply every term by the common denominator to clear the fractions, then solve what remains - often a quadratic. Solving + 4 = x + 6 (multiply by x): 3 + 4x = x² + 6x, so x² + 2x − 3 = 0 and x = 1 or x = −3. Discard any root that makes an original denominator zero.
Practice
Try a Algebraic Fractions question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Simplify .
- x − 3
- x + 3
- x − 9
- x + 9
Show the answer and the working
Answer: x − 3
x² − 9 is a difference of two squares, (x + 3)(x − 3), so the (x + 3) cancels.
- The numerator is a difference of two squares: x² − 9 = (x + 3)(x − 3).
- The fraction becomes .
- Cancel the (x + 3), leaving x − 3.