Expanding and Factorising Quadratics

Welcome to Expanding and Factorising!

Expanding and factorising are inverse operations — two of the most fundamental algebraic skills. Expanding turns brackets into polynomials; factorising reverses the process. Together they unlock quadratic equations, curve sketching, and much more at IGCSE and beyond.

(x + a)(x + b) = x² + (a+b)x + ab

Single Brackets

Distribute over terms

FOIL / Double Brackets

(x+a)(x+b) = x²+…+ab

Special Products

(a+b)², (a−b)², (a+b)(a−b)

Factorising Quadratics

Find two numbers × to c, + to b

Difference of Two Squares

a²−b² = (a+b)(a−b)

1. Expanding Single Brackets

Multiply every term inside the bracket by the term outside.

k(ax + b) = kax + kb

3(x + 5) = 3x + 15
−2(x − 4) = −2x + 8
x(x + 7) = x² + 7x

Watch out for negative multipliers: −2(x − 4) = −2×x + (−2)×(−4) = −2x + 8. Two negatives make a positive.

2. Expanding Double Brackets — FOIL

FOIL stands for First, Outer, Inner, Last. Multiply each term in the first bracket by each term in the second.

(x + a)(x + b) = x² + bx + ax + ab = x² + (a+b)x + ab

Example: (x + 3)(x + 4)
F: x × x = x²
O: x × 4 = 4x
I: 3 × x = 3x
L: 3 × 4 = 12
Sum: x² + 4x + 3x + 12 = x² + 7x + 12

3. Special Products

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

(a + b)(a − b) = a² − b² (Difference of Two Squares)

(x + 5)² = x² + 10x + 25
(x − 3)² = x² − 6x + 9
(x + 4)(x − 4) = x² − 16

4. Factorising x² + bx + c

To factorise x² + bx + c, find two integers p and q such that p + q = b and p × q = c. Then write (x + p)(x + q).

Factorise x² + 7x + 12:
Need two numbers: product = 12, sum = 7
Try 3 and 4: 3 × 4 = 12 ✓ 3 + 4 = 7 ✓
Answer: (x + 3)(x + 4)

Factorise x² − 2x − 8:
Need: product = −8, sum = −2
Try −4 and 2: −4 × 2 = −8 ✓ −4 + 2 = −2 ✓
Answer: (x − 4)(x + 2)

Systematic approach: list all factor pairs of c, then check which pair has the correct sum. For negative c, one factor must be negative.

5. Difference of Two Squares

a² − b² = (a + b)(a − b)

x² − 9 = (x + 3)(x − 3)
x² − 25 = (x + 5)(x − 5)
4x² − 49 = (2x + 7)(2x − 7)

Difference of two squares only works when: (1) both terms are perfect squares, and (2) there is a minus sign between them. You can spot it instantly once you recognise the pattern.

Example 1 — Expand 3(2x + 4)

3 × 2x = 6x 3 × 4 = 12

Answer: 6x + 12

Example 2 — Expand (x + 3)(x + 4)

F: x² O: 4x I: 3x L: 12

Answer: x² + 7x + 12

Example 3 — Expand (x − 5)(x + 2)

F: x² O: 2x I: −5x L: −10

Answer: x² − 3x − 10

Example 4 — Factorise x² + 8x + 15

Need: p × q = 15, p + q = 8. Try 3 and 5: 3 × 5 = 15 ✓, 3 + 5 = 8 ✓

Answer: (x + 3)(x + 5)

Example 5 — Difference of Two Squares: x² − 36

x² − 36 = x² − 6²

Answer: (x + 6)(x − 6)

Example 6 — Expand (x + 3)²

(a + b)² = a² + 2ab + b²: x² + 6x + 9

Answer: x² + 6x + 9

Algebra Tile Model — (x + a)(x + b)

Adjust a and b to see the tile grid representing (x+a)(x+b). The tiles show x², x-terms and the constant.

a = b =

Click Update to see the expansion.

Quadratic Factoriser

Enter b and c for x² + bx + c to find the factorised form.

b = c =

Result appears here.

Exercise 1 — Expanding Single Brackets

Expand each expression. Give the coefficient of x in the result.

1. 3(x + 2). Coefficient of x?

2. 5(x + 1). Coefficient of x?

3. −2(x + 3). Coefficient of x?

4. 4(x − 1). Coefficient of x?

5. −6(x + 2). Coefficient of x?

6. 7(x − 3). Coefficient of x?

7. −3(x − 4). Coefficient of x?

8. 8(x + 1). Coefficient of x?

9. −5(x − 2). Coefficient of x?

10. 9(x − 1). Coefficient of x?

Exercise 2 — Expanding Double Brackets (Constant Term)

Expand each expression. Give the constant term (the number with no x).

1. (x + 2)(x + 3). Constant term?

2. (x + 2)(x + 5). Constant term?

3. (x − 2)(x + 4). Constant term?

4. (x − 3)(x + 5). Constant term?

5. (x + 3)(x + 4). Constant term?

6. (x + 2)(x − 3). Constant term?

7. (x + 4)(x + 5). Constant term?

8. (x − 3)(x + 4). Constant term?

9. (x + 3)(x + 5). Constant term?

10. (x − 4)(x + 5). Constant term?

Exercise 3 — Expanding Double Brackets (x Coefficient)

Expand each expression. Give the coefficient of x (the number multiplying x).

1. (x + 2)(x + 3). Coefficient of x?

2. (x + 3)(x + 4). Coefficient of x?

3. (x + 3)(x − 4). Coefficient of x?

4. (x − 5)(x + 3). Coefficient of x?

5. (x + 5)(x + 3). Coefficient of x?

6. (x − 7)(x + 3). Coefficient of x?

7. (x + 5)(x + 4). Coefficient of x?

8. (x − 8)(x + 3). Coefficient of x?

9. (x + 3)(x + 3). Coefficient of x?

10. (x − 5)(x + 2). Coefficient of x?

Exercise 4 — Factorising Quadratics

Factorise each quadratic into (x + p)(x + q). Give the positive value of p or q (the positive root).

1. Factorise x² + 7x + 12 = (x + ?)(x + 4). Give the missing factor.

2. Factorise x² + 8x + 15 = (x + ?)(x + 3). Give the missing factor.

3. Factorise x² + 8x + 16 = (x + ?)(x + 4). Give the missing factor.

4. Factorise x² + 11x + 28 = (x + ?)(x + 4). Give the missing factor.

5. Factorise x² + 11x + 30 = (x + ?)(x + 5). Give the missing factor.

6. Factorise x² + 5x + 6 = (x + ?)(x + 3). Give the missing factor.

7. Factorise x² + 13x + 40 = (x + ?)(x + 5). Give the missing factor.

8. Factorise x² + 3x + 2 = (x + ?)(x + 2). Give the missing factor.

9. Factorise x² + 14x + 45 = (x + ?)(x + 5). Give the missing factor.

10. Factorise x² + 17x + 70 = (x + ?)(x + 7). Give the missing factor.

Exercise 5 — Difference of Two Squares

Write x² − a² = (x − a)(x + a). Give the value of a.

1. Factorise x² − 9. Value of a?

2. Factorise x² − 25. Value of a?

3. Factorise x² − 16. Value of a?

4. Factorise x² − 49. Value of a?

5. Factorise x² − 36. Value of a?

6. Factorise x² − 4. Value of a?

7. Factorise x² − 64. Value of a?

8. Factorise x² − 100. Value of a?

9. Factorise x² − 81. Value of a?

10. Factorise x² − 1. Value of a?

Practice — 20 Questions

Mixed practice on expanding and factorising.