(x + 6)(x − 6) = x^2 − □

36. A sum times a difference is a difference of squares: −6x and +6x cancel, leaving x^2 − 6^2 = x^2 − 36.

(4x + 3y)(4x − 3y) = □x^2 − □y^2 Fill in each blank

16, 9. A difference of squares: (4x)^2 − (3y)^2 = 16x^2 − 9y^2.

(6x + 7)(7 − 6x) = ? A. 49 − 36x^2 B. 36x^2 − 49 C. 49 + 36x^2 D. 49 − 36x^2 + 84x Choose the right answer

A. 49 − 36x^2. Reorder first: (7 + 6x)(7 − 6x) = 7^2 − (6x)^2 = 49 − 36x^2.

Polynomials & identities · Ages 13–14

Expanding the difference of two squares

Sum times difference = difference of squares: the middle cancels.

How to do it

Method

Sum times difference = difference of squares: the middle cancels

(2x + 5)(2x − 5) = 4x² − 25

Another way

Every term by every term

4x² − 10x + 10x − 25 = 4x² − 25

In full

(a + b)(a − b) ≡ a² − b²: the brackets differ only in the sign between, so multiplied out the middle terms −ab and +ab cancel, leaving the first squared minus the second squared. (2x + 5)(2x − 5) = (2x)² − 5² = 4x² − 25. Check first that both brackets have the same two terms, one added, one taken away; in (2x + 5)(5 − 2x) the first terms differ, so reorder: (5 + 2x)(5 − 2x) = 25 − 4x². Or multiply every term by every term, and see −10x and +10x cancel. The answer is the same. A quick sum: 98 × 102 = (100 − 2)(100 + 2) = 10000 − 4 = 9996.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
(x + 6)(x − 6) = x² −
Show the answer and working
Answer: 36
A sum times a difference is a difference of squares: −6x and +6x cancel, leaving x² − 6² = x² − 36.
Example 2
(4x + 3y)(4x − 3y) = x² − y²
Fill in each blank
Show the answer and working
Answer: 16, 9
A difference of squares: (4x)² − (3y)² = 16x² − 9y².
Example 3
(6x + 7)(7 − 6x) = ?
- A. 49 − 36x²
- B. 36x² − 49
- C. 49 + 36x²
- D. 49 − 36x² + 84x
Choose the right answer
Show the answer and working
Answer: A. 49 − 36x²
Reorder first: (7 + 6x)(7 − 6x) = 7² − (6x)² = 49 − 36x².

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.(x + 14)(x − 14) = x² −
Answer:
2.(2x + 7)(2x − 7) = ?
- A. 2x² − 49
- B. 4x² + 49
- C. 4x² − 28x − 49
- D. 4x² − 49
Choose the right answer
Answer:
3.Work it out quickly with the difference of two squares: 56 × 64 =
Answer:
4.(6x + 5y)(6x − 5y) = x² − y²
Fill in each blank
Answer:
5.(5x + 5)(5 − 5x) = ?
- A. 25 − 25x²
- B. 25 − 25x² + 50x
- C. 25x² − 25
- D. 25 + 25x²
Choose the right answer
Answer:

All the answers

1. 196
2. D. 4x² − 49
3. 3584
4. 36, 25
5. A. 25 − 25x²

Free worksheet: Expanding the difference of two squaresWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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