Practice Maths

Expanding and Factorising Quadratics

Key Ideas

Key Terms

quadratic expression
Has degree 2: the highest power of the variable is 2. General form: ax² + bx + c.
Expanding
Means removing brackets. Use FOIL (First, Outer, Inner, Last) for two binomials.
Factorising
The reverse: write the expression as a product of simpler factors.
HCF first
Before factorising the remaining expression.
Monic quadratic
Leading coefficient = 1 (e.g. x² + 5x + 6). Factorise by finding two numbers that multiply to c and add to b.
Non-monic
Leading coefficient ≠ 1 (e.g. 2x² + 5x + 3). Use the product-sum method or trial and error.
TypeIdentityExample
FOIL(a+b)(c+d) = ac+ad+bc+bd(x+2)(x+5) = x²+7x+10
Perfect square (+)(a+b)² = a²+2ab+b²(x+3)² = x²+6x+9
Perfect square (−)(a−b)² = a²−2ab+b²(x−4)² = x²−8x+16
Diff. of squares(a+b)(a−b) = a²−b²(x+5)(x−5) = x²−25
Monic factorisex²+bx+c = (x+p)(x+q), p×q=c, p+q=bx²+7x+12 = (x+3)(x+4)
Hot Tip — Factorising Strategy Always follow this order: (1) Take out the HCF. (2) Count the terms — two terms? Check for difference of two squares. Three terms? Check for perfect square, then monic, then non-monic. Never skip step 1!

Worked Example — Expanding

Expand (3x − 2)(2x + 5).

First: 3x × 2x = 6x²   Outer: 3x × 5 = 15x   Inner: −2 × 2x = −4x   Last: −2 × 5 = −10

= 6x² + 15x − 4x − 10 = 6x² + 11x − 10

Worked Example — Factorising (non-monic)

Factorise 2x² + 7x + 3.

Product = 2 × 3 = 6. Find two numbers that multiply to 6 and add to 7: 1 and 6.

Split middle term: 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

What Is a Quadratic Expression?

A quadratic expression is any algebraic expression where the highest power of the variable is 2. The general form is ax² + bx + c, where a ≠ 0. Examples include x² + 5x + 6, 3x² − 2x + 1, and x² − 16. When a = 1, the quadratic is called monic; when a ≠ 1, it is non-monic.

Quadratic expressions appear everywhere in mathematics: they describe the path of a thrown ball, the area of rectangles, and the shape of bridges and satellite dishes (parabolas). Being able to expand and factorise them fluently is one of the most important skills in Year 10 algebra.

Expanding Two Brackets Using FOIL

To expand two binomials, multiply every term in the first bracket by every term in the second bracket. The FOIL acronym helps you remember the order: First, Outer, Inner, Last.

For example, (x + 3)(x + 5): F = x², O = 5x, I = 3x, L = 15. Collecting like terms: x² + 8x + 15.

For (2x − 1)(x + 4): F = 2x², O = 8x, I = −x, L = −4. Result: 2x² + 7x − 4. Always check your signs carefully — this is where most errors occur.

Special Products: Worth Memorising

Three binomial products appear so often in mathematics that they deserve their own identities. Memorising these saves time and prevents errors on exams.

Perfect square (sum): (a + b)² = a² + 2ab + b². The middle term is twice the product of the two terms. Students often forget the middle term and incorrectly write (a + b)² = a² + b² — this is wrong!

Perfect square (difference): (a − b)² = a² − 2ab + b². Note that the last term b² is always positive.

Difference of two squares: (a + b)(a − b) = a² − b². This is remarkable: expanding gives only two terms, with no middle term. This identity is incredibly useful for mental arithmetic: 101 × 99 = (100+1)(100−1) = 10000 − 1 = 9999.

Factorising: The Reverse Process

Factorising means writing an expression as a product of factors. It is the reverse of expanding. The general strategy is:

Step 1: Always take out the Highest Common Factor (HCF) first. For example, 6x² + 9x = 3x(2x + 3). This makes the remaining factorisation simpler.

Step 2: After removing the HCF, look at the remaining expression. If it has two terms, check if it is a difference of two squares. If it has three terms, try factorising as a trinomial.

Monic trinomials (a = 1): For x² + bx + c, find two integers p and q where p × q = c and p + q = b. Then x² + bx + c = (x + p)(x + q). For example, to factorise x² − 7x + 12: find p × q = 12 and p + q = −7. That’s p = −3, q = −4. So x² − 7x + 12 = (x − 3)(x − 4).

Non-monic trinomials (a ≠ 1): For ax² + bx + c, find two numbers that multiply to ac and add to b. Use these to split the middle term, then factorise by grouping. For 2x² + 5x + 3: ac = 6, find 2 and 3 (2×3=6, 2+3=5). Split: 2x² + 2x + 3x + 3 = 2x(x+1) + 3(x+1) = (2x+3)(x+1).

Common exam mistake: Students often miss the HCF before factorising a trinomial. For example, 2x² − 2x − 12 should first become 2(x² − x − 6), then factorise to 2(x − 3)(x + 2). Writing (2x + 4)(x − 3) or similar without factoring out the 2 is technically correct but incomplete — always write factors in their simplest form.

Mastery Practice

  1. Expand each product using FOIL. Fluency

     ExpressionExpanded form
    (a)(x + 4)(x + 2) 
    (b)(x − 5)(x + 3) 
    (c)(2x + 1)(x + 4) 
    (d)(3x − 2)(2x + 5) 
    (e)(x + 2y)(x − 3y) 
    (f)(4 − x)(3 + 2x) 

  2. Expand each expression using the appropriate special product identity. Fluency

     ExpressionIdentity used & result
    (a)(x + 6)² 
    (b)(2x − 3)² 
    (c)(x + 8)(x − 8) 
    (d)(5x + 2)(5x − 2) 
    (e)(3x + y)² 
    (f)(1 − 4x)² 

  3. Factorise each monic quadratic completely. Fluency

     ExpressionFactorised form
    (a)x² + 7x + 12 
    (b)x² − 8x + 15 
    (c)x² + x − 20 
    (d)x² − 2x − 35 
    (e)x² − 10x + 25 
    (f)x² − 49 

  4. Take out the highest common factor first, then factorise the remaining expression completely. Fluency

     ExpressionFully factorised
    (a)3x² + 9x 
    (b)2x² − 18 
    (c)4x² − 8x − 12 
    (d)5x² − 20x + 20 

  5. Factorise each non-monic quadratic. Use the product-sum method or trial and error. Understanding

    Product-Sum Method. For ax² + bx + c, find two numbers with product = ac and sum = b. Split the middle term and factorise by grouping.
    1. 2x² + 5x + 2
    2. 3x² − 7x + 2
    3. 4x² + 12x + 9
    4. 6x² + x − 2
    5. 2x² − x − 10

  6. Factorise each expression fully using the difference of two squares or other appropriate technique. Understanding

    1. 4x² − 25
    2. 9x² − y²
    3. 16a² − 81b²
    4. x4 − 81   (Hint: factorise twice)
    5. 3x² − 75

  7. Rectangle dimensions. Understanding

    Garden Design. A rectangular garden has length (x + 5) metres and width (x + 2) metres.
    1. Write an expression for the area by expanding the product.
    2. Given that the area of the garden is 40 m², form an equation and solve for x (where x > 0).
    3. State the actual dimensions and calculate the perimeter of the garden.

  8. Special products in context. Understanding

    1. Use the difference of two squares identity to calculate 103 × 97 without a calculator. Show your reasoning clearly.
    2. Expand and simplify (x + 5)² + (x − 5)². What do you notice about the x-term in your answer?
    3. Factorise 4x² − 20x + 25 and identify the special product type.
    4. Factorise 3x² − 3 completely, showing all steps.

  9. Expand and simplify. Problem Solving

    Multi-step simplification. Expand fully and collect like terms.
    1. Expand and simplify (x + 2)² − (x − 4)(x + 3).
    2. For what value of x does (x + 2)² − (x − 4)(x + 3) = 36?
    3. Show algebraically that (x + 2)² − (x − 4)(x + 3) is always positive when x > −3.2.

  10. Algebraic proof. Problem Solving

    Proof using algebra. Use expansion identities to prove the following results.
    1. Prove that (a + b)² − (a − b)² = 4ab.
    2. Use the result from part (a) to evaluate 75² − 25² without a calculator.
    3. Prove that the difference of the squares of two consecutive even integers is always divisible by 4. Let the integers be 2n and 2n + 2.
See Answers ➔