Algebraic Fractions – Grade 10 IGCSE Maths

Welcome to Algebraic Fractions!

Algebraic fractions are fractions where the numerator and/or denominator contain algebraic expressions. They appear throughout IGCSE algebra — in simplification, equation solving, and problem modelling. The key insight: you can only cancel factors, never terms.

Simplify by factorising first | Find LCM to add/subtract | Multiply both sides by LCM to solve equations

Factorise numerator and denominator, then cancel common factors
Add and subtract algebraic fractions with different denominators
Multiply and divide algebraic fractions
Solve equations that contain algebraic fractions
Handle equations leading to quadratics

Simplifying

Factorise first, then cancel factors

Adding & Subtracting

LCM of denominators

Multiplying & Dividing

Multiply across, flip and multiply

Solving Equations

Multiply by LCM to clear fractions

Quadratic Results

Equations that lead to ax²+bx+c=0

Common Pitfalls

Cancelling terms vs factors

Learn 1 — Simplifying Algebraic Fractions

To simplify an algebraic fraction: factorise the top and bottom completely, then cancel any common factors. You can only cancel something that multiplies the entire numerator AND multiplies the entire denominator.

Step 1: Factorise numerator | Step 2: Factorise denominator | Step 3: Cancel common factors

Example 1: Simplify (x² − 4) / (x + 2)
Numerator: x² − 4 = (x + 2)(x − 2) [difference of two squares]
Denominator: (x + 2) — already factorised
Cancel (x + 2): (x + 2)(x − 2) / (x + 2) = x − 2
(Valid for all x ≠ −2)

Example 2: Simplify (2x² + 5x + 3) / (x² − 1)
Numerator: 2x² + 5x + 3 = (2x + 3)(x + 1)
Denominator: x² − 1 = (x + 1)(x − 1)
Cancel (x + 1): (2x + 3) / (x − 1)

Example 3: Simplify (3x² − 12) / (6x + 12)
Numerator: 3(x² − 4) = 3(x + 2)(x − 2)
Denominator: 6(x + 2)
Cancel 3 and (x + 2): (x − 2) / 2

NEVER cancel individual terms. For example:
(x² + 3x) / (x + 3) ≠ x [WRONG — you cannot cancel 3x and 3]
Correct: x(x + 3) / (x + 3) = x [factorise first, then cancel (x+3)]

Multiplying and Dividing Algebraic Fractions

Multiply: (a/b) × (c/d) = ac/bd — cross-cancel before multiplying if possible.
Example: [(x+2)/(x−3)] × [(x²−9)/(x+2)] = [(x+2)/(x−3)] × [(x+3)(x−3)/(x+2)] = x+3

Divide: (a/b) ÷ (c/d) = (a/b) × (d/c) — flip the second fraction, then multiply.
Example: (x²−4)/(x+1) ÷ (x−2)/(2x+2) = [(x+2)(x−2)/(x+1)] × [2(x+1)/(x−2)] = 2(x+2)

Learn 2 — Adding and Subtracting Algebraic Fractions

Just like numeric fractions — find a common denominator, convert each fraction, then add or subtract the numerators. The tricky part is finding the LCM of algebraic denominators, which requires factorising first.

LCM of denominators → rewrite fractions → add/subtract numerators → simplify

Simple cases

Example: 2/x − 3/x²
LCM of x and x² is x²
2/x = 2x/x² and 3/x² stays as 3/x²
Result: (2x − 3)/x²

Example: 3/(x + 2) + 2/(x − 1)
LCM = (x + 2)(x − 1)
3/(x+2) = 3(x−1)/[(x+2)(x−1)]
2/(x−1) = 2(x+2)/[(x+2)(x−1)]
Sum = [3(x−1) + 2(x+2)] / [(x+2)(x−1)]
Numerator: 3x − 3 + 2x + 4 = 5x + 1
Result: (5x + 1) / [(x+2)(x−1)]

When you need to factorise denominators

Example: 2/(x² − 4) + 3/(x + 2)
Factorise: x² − 4 = (x+2)(x−2)
LCM = (x+2)(x−2) — already contains (x+2) as a factor
2/[(x+2)(x−2)] stays the same
3/(x+2) = 3(x−2)/[(x+2)(x−2)]
Sum = [2 + 3(x−2)] / [(x+2)(x−2)] = [2 + 3x − 6] / [(x+2)(x−2)] = (3x − 4) / [(x+2)(x−2)]

Always factorise both denominators before finding the LCM. The LCM uses each distinct factor at its highest power — like finding LCM of numbers, but with expressions.

Learn 3 — Equations with Algebraic Fractions

When an equation contains algebraic fractions, multiply every term on both sides by the LCM of all denominators. This clears all fractions in one step, leaving a polynomial equation to solve.

Multiply every term by LCM → expand carefully → collect terms → solve

Example: 3/(x − 1) + 2/(x + 2) = 1
LCM = (x−1)(x+2)
Multiply every term: 3(x+2) + 2(x−1) = 1×(x−1)(x+2)
Left side: 3x + 6 + 2x − 2 = 5x + 4
Right side: x² + x − 2
Rearrange: 0 = x² + x − 2 − 5x − 4 = x² − 4x − 6
Solve: x = (4 ± √(16 + 24))/2 = (4 ± √40)/2 = 2 ± √10

Example: x/(x+1) = 3/4
Cross-multiply (LCM = 4(x+1)): 4x = 3(x+1) = 3x + 3
x = 3

Example: 2/(x−3) − 1/x = 1/2
LCM = 2x(x−3)
Multiply every term by 2x(x−3):
2 × 2x = 4x
−1 × 2(x−3) = −2(x−3) = −2x + 6
right: 1 × x(x−3)
4x − 2x + 6 = x² − 3x
2x + 6 = x² − 3x
x² − 5x − 6 = 0
(x−6)(x+1) = 0 → x = 6 or x = −1

After solving, check that your solutions don't make any denominator equal to zero. If a solution makes a denominator 0, it must be rejected.

Example 1 — Simplify with factorising (difference of squares)

Q: Simplify (x² − 9) / (2x + 6)

M1: Factorise numerator: x² − 9 = (x+3)(x−3)

M1: Factorise denominator: 2x + 6 = 2(x+3)

A1: Cancel (x+3): (x−3)/2 A1

Example 2 — Simplify with quadratic numerator

Q: Simplify (3x² − x − 2) / (x² − 1)

M1: Numerator: 3x² − x − 2 = (3x + 2)(x − 1)

M1: Denominator: x² − 1 = (x+1)(x−1)

A1: Cancel (x−1): (3x+2)/(x+1) A1

Example 3 — Add fractions with different denominators

Q: Write as a single fraction: 4/(x+3) − 2/(x−1)

M1: LCM = (x+3)(x−1)

M1: 4(x−1)/[(x+3)(x−1)] − 2(x+3)/[(x+3)(x−1)]

Numerator: 4x−4 − 2x−6 = 2x − 10

A1: (2x−10) / [(x+3)(x−1)] = 2(x−5)/[(x+3)(x−1)] A1

Example 4 — Solve equation leading to quadratic

Q: Solve 5/(x+2) − 1/(x−1) = 2

M1: LCM = (x+2)(x−1). Multiply every term.

M1: 5(x−1) − 1(x+2) = 2(x+2)(x−1)

Left: 5x − 5 − x − 2 = 4x − 7

Right: 2(x²+x−2) = 2x²+2x−4

M1: 0 = 2x²+2x−4−4x+7 = 2x²−2x+3
Discriminant: 4−24 = −20 < 0 → No real solutions A1

Example 5 — Multiply algebraic fractions

Q: Simplify [(x²+3x)/(x²−4)] × [(x+2)/(x+3)]

M1: Factorise: x²+3x = x(x+3) | x²−4 = (x+2)(x−2)

M1: [x(x+3)/((x+2)(x−2))] × [(x+2)/(x+3)]

A1: Cancel (x+3) and (x+2): x/(x−2) A1

Example 6 — Solve with three fractions

Q: Solve 3/x + 2/(x+1) = 5

M1: LCM = x(x+1). Multiply every term by x(x+1).

M1: 3(x+1) + 2x = 5x(x+1)

3x+3+2x = 5x²+5x → 5x+3 = 5x²+5x → 5x²=3 → x² = 3/5

A1: x = ±√(3/5) = ±√15/5 A1

Common Mistakes — Algebraic Fractions

Mistake 1 — Cancelling terms instead of factors

✗ Wrong: (x² + 3) / (x + 3) = x (cancelling "+3" from top and bottom)

✓ Correct: x² + 3 cannot be factorised to include (x+3), so it does NOT simplify this way.

You can only cancel something that is a factor of the ENTIRE numerator AND a factor of the ENTIRE denominator. Terms added/subtracted are NOT factors.

Mistake 2 — Not multiplying EVERY term when clearing fractions

✗ Wrong: 2/(x+1) + 3 = 4/(x−1) → multiply by (x+1)(x−1): gives 2(x−1) + 3 = 4(x+1)

✓ Correct: Every term gets multiplied: 2(x−1) + 3(x+1)(x−1) = 4(x+1)

The number 3 (standing alone) must also be multiplied by the LCM. Students often forget non-fraction terms.

Mistake 3 — Sign errors when expanding after multiplying through

✗ Wrong: −2/(x−3) × (x−3) = +2 (forgetting the negative sign)

✓ Correct: −2/(x−3) × (x−3) = −2

When multiplying to clear fractions, carry the sign of the entire fraction term through to the numerator. Bracket generously: −[2(x+1)].

Mistake 4 — Not factorising fully before simplifying

✗ Wrong: (2x + 4)/(x + 2) → student sees (x+2) in denominator but numerator is "2x+4" not factorised → gives up

✓ Correct: 2x+4 = 2(x+2), so (2x+4)/(x+2) = 2(x+2)/(x+2) = 2

Always take out numerical factors too — factorise the numerator and denominator completely before looking for common factors.

Mistake 5 — Losing a solution by cancelling a factor that could be zero

✗ Wrong: Solving x(x−2)/(x−2) = 3 → cancel (x−2) → x = 3, forgetting to check x = 2

✓ Correct: x = 2 makes denominator = 0, so x = 2 is not in the domain. x = 3 is the only valid solution. (In this case the answer is the same, but be aware of domain restrictions.)

Key Formulas — Algebraic Fractions

Given in Exam	Must Memorise / Apply
Expression to simplify	Factorise numerator & denominator; cancel common factors only
—	a/b + c/d = (ad + bc)/bd (find LCM first for efficiency)
—	a/b × c/d = ac/bd (cross-cancel if possible)
—	a/b ÷ c/d = a/b × d/c (KCF: Keep Change Flip)
—	To solve: multiply all terms by LCM of denominators
—	x² − a² = (x+a)(x−a) [very common in denominators]
—	ax² + bx + c → factorise by inspection or quadratic formula

Golden rule: You can only cancel FACTORS — things that multiply the WHOLE expression

After solving, always check solutions don't make any denominator = 0

Common Factorisations You'll Need

x² − 1 = (x+1)(x−1)
x² − 4 = (x+2)(x−2)
x² − 9 = (x+3)(x−3)
x² + 2x + 1 = (x+1)²
x² − 2x + 1 = (x−1)²
2x² + 5x + 3 = (2x+3)(x+1)
3x² + x − 2 = (3x − 2)(x + 1)

Algebraic Fraction Simplifier

Enter a quadratic-over-linear or quadratic-over-quadratic fraction (up to degree 2). Provide coefficients and the simplifier will factor and show cancellation.

Numerator: ax² + bx + c | Denominator: dx² + ex + f

Num a: b: c:

Den d: e: f:

Enter coefficients and click Simplify.
Example: numerator x²−x−6, denominator x²−4x+3

Exercise 1 — Simplify (factorise first)

Enter the simplified numerator (e.g. for (x−2)/(3), enter numerator = x−2, enter the number that comes out. For these questions enter just the key value or expression label as prompted.)

1. Simplify (x²−25)/(x+5). What is the simplified expression? (It equals x + __)

2. Simplify (2x+6)/(x+3). What does it simplify to?

3. Simplify (x²−x)/(x²+2x−3). Answer is (x)/(x+?)

4. Simplify (x²+5x+6)/(x+3). Answer is x + ?

5. Simplify (4x²−16)/(2x+4). Answer = 2(x − ?)

6. Simplify (x²+x−12)/(x−3). Answer = x + ?

7. Simplify (2x²−8x)/(x²−16). Answer = 2x/(x+?)

8. Simplify (3x²+9x)/(x²+6x+9). Answer = 3x/(x+?)

Exercise 2 — Add/Subtract (evaluate at a specific x for checking)

Evaluate each combined expression at x = 4 and enter the decimal answer.

1. 1/(x+2) + 1/(x−2) at x=4. (Hint: combined = 2x/(x²−4))

2. 3/(x+1) − 2/(x+2) at x=4. (Hint: combined = (x+4)/[(x+1)(x+2)])

3. 2/x + 3/x² at x=4. (Hint: combined = (2x+3)/x²)

4. 5/(x−3) − 1/(x+1) at x=4. (Hint: combined = (4x+8)/[(x−3)(x+1)])

5. 1/(x+2) + 2/(x²−4) at x=4. (Hint: combined = (x−2+2)/[(x+2)(x−2)] = 1/(x−2))

6. 4/(x+3) + 2/(x−3) at x=5. (Hint: combined = (6x−6)/[(x+3)(x−3)])

7. 3/(x+1) + 3/(x−1) at x=4. (Hint: combined = 6x/(x²−1))

8. 2/(x+5) − 1/(x−5) at x=6. (Hint: combined = (x−15)/[(x+5)(x−5)])

Exercise 3 — Solve equations with algebraic fractions

1. Solve 3/(x+1) = 6/(x+4). Find x.

2. Solve x/(x+2) = 4/5. Find x.

3. Solve 2/(x−1) + 1/x = 1. Find the positive solution.

4. Solve 4/x − 1/(x+1) = 1. Find the positive solution.

5. Solve 3/(x+2) = 2/(x−1). Find x.

6. Solve 1/(x−2) − 1/(x+2) = 1. Find the positive solution (round to 2dp).

7. Solve 6/(x+1) + 2/(x−1) = 1. Enter the larger solution (1dp).

8. Solve 10/x = x + 3. Find the positive solution.

Exercise 4 — Multiply and Divide Algebraic Fractions (evaluate at x=3)

Simplify each expression, then evaluate at x = 3.

1. [(x+2)/(x−1)] × [(x²−1)/(x+2)] — evaluate at x=3.

2. [(x²−4)/(x+3)] × [(x+3)/(x−2)] — evaluate at x=3.

3. [(x²+2x)/(x+1)] ÷ [x/(x+1)] — evaluate at x=3.

4. [(x²−9)/(2x)] ÷ [(x−3)/4] — evaluate at x=5.

5. [(2x+4)/(x²−1)] × [(x−1)/2] — evaluate at x=3.

6. [(x²+x−6)/(x+3)] × [1/(x−2)] — evaluate at x=4.

7. [(x+5)/(x²−25)] ÷ [1/(x−5)] — evaluate at x=6.

8. [(3x²−3)/(x+1)] ÷ [3(x−1)] — evaluate at x=2.

Exercise 5 — Mixed Harder Problems

1. Simplify (x²−x−6)/(x²+x−12). Answer = (x+a)/(x+b). Enter a+b.

2. Write as a single fraction: 2/(x+1) − 3/(x−2). Evaluate at x=5. (answer to 3dp)

3. Solve: 2/(x+3) + 1/(x−1) = 1. Find the larger root (2dp).

4. Simplify [(x²−4x+4)/(x²−4)] — what does x cancel from, leaving (x+?)/(1)?
Answer = (x−2)/(x+?). Enter the number.

5. Solve 1/(x−1) + 1/(x+1) = 1. Find the positive solution (2dp).

6. Simplify [(x²+6x+9)/(x²+2x−3)] — answer = (x+3)/(x+?). Enter the number.

7. Simplify [(2x²+x−6)/(x²−4)] — answer = (2x−3)/(x−?). Enter the number.

8. Solve 3/(x+2) − 2/(x−2) = 1/(x²−4). Find x.

Practice — 25 Mixed Questions

🔵 = Non-calculator 🟢 = Calculator allowed

🔵 1. Simplify (x²−16)/(x+4). Answer = x − ?

🔵 2. Simplify (3x−9)/(x²−9). Numerator of answer = 3, denom = x + ?

🔵 3. Evaluate (x+3)/(x−1) at x=5. Answer?

🟢 4. Solve 6/(x+1) = 3. Find x.

🟢 5. 1/(x+3) + 1/(x+3) = ? (single fraction, evaluate at x=2)

🔵 6. Simplify (5x²−5)/(x+1). Answer = 5(x−?)

🟢 7. Solve x/(x−2) = 3. Find x.

🔵 8. Simplify (x²+2x+1)/(x+1). Answer = x + ?

🟢 9. Evaluate 2/(x−3) + 1/(x+3) as single fraction at x=5. (3/(x+3) + ?) Evaluate at x=5.

🔵 10. Simplify (2x²+4x)/(x+2). Answer = 2x

🟢 11. Solve 2/x − 1 = 3/4. Find x.

🔵 12. Simplify (x²−3x)/(x²−9). Answer = x/(x+?)

🟢 13. Solve 4/(x+2) + 1/(x−2) = 2. Find the positive solution (2dp).

🔵 14. Evaluate [(x+1)/(x−1)] × [(x−1)/(x+2)] at x=4.

🟢 15. Solve 5/x = x − 4. Find positive x.

🔵 16. Simplify (x²−4x+4)/(x−2). Answer = x − ?

🟢 17. Evaluate 3/(x+1) − 2/(x−1) as single fraction at x=3.

🔵 18. Simplify (4x+8)/(x²−4). Answer = 4/(x − ?)

🟢 19. Solve 3/(x+2) = 2/(x−1). Find x.

🔵 20. [(x−3)/(x+1)] × [(x+1)/(x+2)] at x=5.

🟢 21. Solve 1/(x+1) + 1/(x+2) = 1/2. Find the positive solution (2dp).

🔵 22. Simplify (6x²−6)/(3x+3). Answer = 2(x − ?)

🟢 23. Evaluate (x²+x−2)/(x−1) at x=6.

🟢 24. Solve 2/(x−3) = x/4. Find the positive solution.

🔵 25. Simplify (2x²−8)/(x²+2x−8). Answer = (2(x+?))/((x+?)). Enter first factor's number.

Challenge — 12 Questions (IGCSE Extended Level)

1. Simplify (x³−x)/(x²−1). Evaluate at x=5.

2. Solve 3/(x+1) − 2/(x−1) = 1/(x²−1). Find x.

3. Show (x²+4x+4)/(x²−4) = (x+2)/(x−2). Evaluate at x=6.

4. Simplify [1/(x−2) − 1/(x+2)] / [(x²−4)/4]. This simplifies to a/(x²−4)². Enter a.

5. Solve: (x+2)/(x−1) − (x−2)/(x+1) = 1. Find exact value of x.

6. Solve 4/(x²−9) = 1/(x−3) + 1/(x+3). Find x (if no real solution, enter 999).

7. Simplify [(x²+x−2)/(x²−x−6)] × [(x²−4x+3)/(x²−1)]. Evaluate at x=5.

8. Solve x/(x+3) + (x+3)/x = 13/6. Find the positive solution.

9. Simplify (2x²−5x+2)/(2x²−x−1). Evaluate at x=4.

10. Write (3x+1)/(x²+x−2) as partial fractions A/(x−1) + B/(x+2). Find A+B.

11. Solve 1/(x−2) + 2/(x+3) = 1. Find the larger solution (2dp).

12. Simplify [(x²−6x+9)/(x²−9)] ÷ [(x−3)/(x+3)]. Evaluate at x=5.

Exam Style Questions

Question 1 — Simplify [4 marks]

Simplify fully: (2x² + x − 6) / (x² − 4)

(a) Factorise the numerator. Enter the product of constant terms in the two factors.

(b) Write down the simplified fraction. It equals (2x−3)/(x−?). Enter the number.

(c) Evaluate your simplified expression at x = 5.

Question 2 — Write as a single fraction [4 marks]

Write (3/(x+2)) − (2/(x−3)) as a single fraction in its simplest form.

(a) State the common denominator.
Enter the product of the two brackets as (x+?)(x−?): enter the two numbers separated — first number:

(b) Find the simplified numerator when the fractions are combined (it is ax+b). Enter a.

(c) Evaluate the single fraction at x = 5.

Question 3 — Solve equation [5 marks]

Solve: 3/(x−2) + 2/(x+1) = 2

(a) Multiply through by the LCM. State the LCM (enter as two factors; enter the value of the LCM at x=4).

(b) After expanding, write the equation in form ax²+bx+c=0. Enter b (the coefficient of x).

(c) Solve the quadratic. Enter the larger solution (2 decimal places).

Question 4 — Show and simplify [4 marks]

Simplify: [(x²+5x+6)/(x²+x−6)] ÷ [(x+2)/(x−2)]

(a) Factorise x²+5x+6. Enter the larger constant factor.

(b) The expression simplifies to 1. Verify by evaluating at x=5 (enter value).

Question 5 — Problem solving [5 marks]

A rectangle has length (x+3)/(x−2) cm and width (x²−4)/(x+3) cm.

(a) Find a simplified expression for the area. It equals (x+?)(x+2)/(x+?) × ... simplify to: x+2 (enter 2 if correct, else enter simplified constant).

(b) Find the area when x = 6.

(c) If the area equals 10, find the value of x.

Algebraic Fractions GCSE Level