Practice Maths

Expanding Binomial Products

Key Ideas

Key Terms

binomial
An algebraic expression with exactly two terms — e.g. (x + 4) or (3x − 2).
perfect square
The result of squaring a binomial — (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
difference of two squares
The identity (a + b)(a − b) = a² − b² — the product of a sum and difference with no middle term.
Identity Formula Key Feature
Perfect square (sum)(a + b)2 = a2 + 2ab + b2Middle term is +2ab
Perfect square (diff)(a − b)2 = a2 − 2ab + b2Last term is always positive
Difference of two squares(a + b)(a − b) = a2 − b2No middle term (+ab − ab = 0)
Hot Tip To check for difference of two squares: are both binomials identical except one has + and the other has −? If yes, the middle terms cancel and you get a2 − b2 directly.

Worked Example — Perfect Square

Question: Expand (3x − 5)2.

Identity: (a − b)2 = a2 − 2ab + b2 with a = 3x, b = 5.

a2 = 9x2  |  2ab = 30x  |  b2 = 25

Result: 9x2 − 30x + 25

Verify (FOIL): (3x − 5)(3x − 5) = 9x2 − 15x − 15x + 25 = 9x2 − 30x + 25 ✓

Why Learn Special Products?

You could always expand binomial products using FOIL, but three special patterns appear so frequently in algebra that it is worth memorising them. Recognising and applying these patterns saves time in exams and reduces the chance of error.

Perfect Square: (a + b)2

(a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2.

The pattern: square the first term, double the product of both terms, square the last term. Example: (x + 5)2 = x2 + 10x + 25. Example: (3x + 2)2 = 9x2 + 12x + 4.

Warning: (a + b)2 is NOT a2 + b2. The middle term 2ab is always there and is very commonly forgotten.

Perfect Square: (a − b)2

(a − b)2 = a2 − 2ab + b2. The only change from the previous pattern is that the middle term is negative.

Example: (x − 4)2 = x2 − 8x + 16. Example: (2x − 3)2 = 4x2 − 12x + 9.

Difference of Two Squares: (a + b)(a − b)

(a + b)(a − b) = a2 − ab + ab − b2 = a2 − b2. The two middle terms cancel exactly, leaving only the difference of the squares. This is why it is called the "difference of two squares."

Example: (x + 7)(x − 7) = x2 − 49. Example: (3x + 5)(3x − 5) = 9x2 − 25.

This pattern is also useful for mental arithmetic: 31 × 29 = (30 + 1)(30 − 1) = 900 − 1 = 899.

Exam tip: The most frequently lost mark in this topic is writing (a + b)2 = a2 + b2. This is wrong — it is called the "freshman's dream" error. Always expand fully. If you are unsure, verify with a number: (2 + 3)2 = 25, but 22 + 32 = 13. They are not equal. The correct expansion (2 + 3)2 = 4 + 12 + 9 = 25 includes the middle term 2 × 2 × 3 = 12.

Mastery Practice

  1. Expand each perfect square using the identity. Fluency

     ExpressionExpanded Form
    (a)(x + 5)2 
    (b)(y − 4)2 
    (c)(2x + 3)2 
    (d)(3y − 7)2 
    (e)(4p − 9)2 
    (f)(6k + 1)2 
    (g)(5m + 4)2 
    (h)(7a − 2)2 

  2. Expand each difference of two squares using (a + b)(a − b) = a2 − b2. Fluency

     ExpressionExpanded Form
    (a)(a + 8)(a − 8) 
    (b)(5m + 2)(5m − 2) 
    (c)(x + 11)(x − 11) 
    (d)(7a − 3)(7a + 3) 
    (e)(3p + 10)(3p − 10) 
    (f)(2x + 9)(2x − 9) 
    (g)(6y + 5)(6y − 5) 
    (h)(4k − 7)(4k + 7) 

  3. Identify the identity type, then expand. Fluency

     ExpressionIdentity TypeExpanded Form
    (a)(x + 2)(x + 9)  
    (b)(3a + 5)2  
    (c)(2m + 7)(2m − 7)  
    (d)(4n − 3)2  
    (e)(5p − 2)(5p + 2)  
    (f)(2p − 9)(3p + 1)  
    (g)(x + 1)2  
    (h)(9x + 4)(9x − 4)  

  4. True or False? Write T or F and correct any false statements. Fluency

     StatementT / F
    (a)(a + b)2 = a2 + b2 
    (b)The last term in (a − b)2 is always positive. 
    (c)(x − 3)2 = (3 − x)2 
    (d)(x + y)(x − y) has no middle (xy) term. 

  5. Expand, then verify by substitution. Understanding

    Checking your work. Expanding and then substituting a value is a powerful way to confirm your answer is correct.
    1. Expand (x + 3)(x − 3). Verify by substituting x = 5 into both the original and your expanded form.
    2. Expand (2a + 1)2. Verify by substituting a = 3.
    3. Expand (4m − 2)(4m + 2). Verify by substituting m = 2.

  6. Using identities for mental arithmetic. Understanding

    Smart calculation. Algebra identities can make large multiplications easier.
    1. Use the difference of two squares to evaluate 49 × 51 without a calculator. (Hint: write as (50 − 1)(50 + 1).)
    2. Use the perfect square identity to evaluate 1022. (Hint: 102 = 100 + 2.)
    3. Use the perfect square identity to evaluate 972. (Hint: 97 = 100 − 3.)

  7. Area expressions with binomial products. Understanding

    Geometry meets algebra. A square garden has side length (x + 4) m. A uniform path of width 1 m surrounds the outside of the garden.
    1. Write an expression for the side length of the outer square (garden + path).
    2. Expand and simplify to find the total area (garden + path).
    3. Write an expression for the area of just the path (total area − garden area).
    4. Simplify your path area expression.

  8. Reasoning with special products. Understanding

    Think it through. Consider the expression (n + 7)2 − (n − 7)2.
    1. Expand (n + 7)2.
    2. Expand (n − 7)2.
    3. Subtract to find the simplified difference. What do you notice about the result?
    4. Is (a + b)2 ever equal to a2 + b2? Under what conditions?

  9. Swimming pool dimensions. Problem Solving

    Pool design. A rectangular swimming pool has dimensions (3x + 2) m by (3x − 2) m.
    1. Write a simplified expression for the area of the pool.
    2. Find the area when x = 5.
    3. Find the perimeter when x = 5.
    4. If the pool area must be at least 100 m2, what is the minimum integer value of x? (Substitute values to test.)

  10. Investigation — expanding beyond two brackets. Problem Solving

    Going further. You can expand three brackets by first expanding any two, then expanding the result with the third.
    1. Expand (x + 1)(x + 2) first.
    2. Now multiply your result by (x + 3) to find (x + 1)(x + 2)(x + 3). Expand fully and collect like terms.
    3. Verify your answer is correct by substituting x = 1 into both the original three-bracket product and your expanded form.
    4. What is the degree (highest power of x) of the final expanded expression?
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