Factorisation · Ages 13–14
Factorising perfect squares
First and last are squares, the middle twice their product.
How to do it
Method
First and last are squares, the middle twice their product
x2 − 10x + 25 = (x − 5)2
Another way
Expand to check
(x − 5)2 = x2 − 10x + 25
In full
Read the perfect squares backwards: a2 + 2ab + b2 ≡ (a + b)2 and a2 − 2ab + b2 ≡ (a − b)2. Check the first and last terms are squares (x2 = x2, 25 = 52), then that the middle term is twice their product: in x2 − 10x + 25, 10x = 2 × x × 5, so it is (x − 5)2, the middle sign going inside the bracket. 4x2 + 12x + 9: 4x2 = (2x)2, 9 = 32 and 12x = 2 × 2x × 3, so it is (2x + 3)2. If the middle isn't twice the product (x2 + 5x + 25, say), it is not a perfect square. Or expand the answer to check. The answer is the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
x2 + 4x + 4 = (x + )2
Show the answer and working
Answer: 2
4 = 22 and the middle 4x = 2 × x × 2, so it is (x + 2)2.
Example 2
Factorise x2 − 4x + 4. What do you get?
- A. (x − 4)2
- B. (x + 2)(x − 2)
- C. (x − 2)2
- D. (x + 2)2
Choose the right answer
Show the answer and working
Answer: C. (x − 2)2
x2 and 4 = 22 are squares and the middle 4x = 2 × x × 2, with a minus: (x − 2)2.
Example 3
25x2 + 70x + 49 = (x + )2
Fill in each blank
Show the answer and working
Answer: 5, 7
25x2 = (5x)2, 49 = 72 and the middle 70x = 2 × 5x × 7, so it is (5x + 7)2.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.x2 + 12x + 144 is a perfect square.
True or false?
Answer:
2.4x2 − 72x + 324 = (x − )2
Fill in each blank
Answer:
3.9x2 + 24x + 16 = (x + )2
Fill in each blank
Answer:
4.9x2 + kx + 25 is a perfect square and k is positive, so k = .
Answer:
5.x2 + 6x + k is a perfect square, so k = .
Answer:
All the answers
- 1. False
- 2. 4, 9
- 3. 3, 4
- 4. 30
- 5. 9
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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