Polynomials & identities · Ages 13–14
Expanding perfect squares
First squared, twice the product, last squared: keep the middle term.
How to do it
Method
First squared, twice the product, last squared: keep the middle term
(3x − 2)2 = 9x2 − 12x + 4
Another way
As two brackets multiplied
(3x − 2)(3x − 2) = 9x2 − 6x − 6x + 4
In full
(a + b)2 ≡ a2 + 2ab + b2 and (a − b)2 ≡ a2 − 2ab + b2: the first term squared, plus or minus twice the two terms' product, plus the last term squared. (3x − 2)2: 3x squared is 9x2, twice the product is 2 × 3x × 2 = 12x, and 2 squared is 4, so it is 9x2 − 12x + 4. The commonest slip is writing a2 + b2 and losing the 2ab in the middle. Or write it as (3x − 2)(3x − 2) and multiply every term by every term. The answer is the same. It makes quick sums too: 1032 = (100 + 3)2 = 10000 + 600 + 9 = 10609.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
(x − 12)2 = ?
- A. x2 − 144
- B. x2 − 24x + 144
- C. x2 − 12x + 144
- D. x2 + 144
Choose the right answer
Show the answer and working
Answer: B. x2 − 24x + 144
(a − b)2 ≡ a2 − 2ab + b2: x2 − 2 × x × 12 + 122 = x2 − 24x + 144.
Example 2
(4x + 8)2 = x2 + x +
Fill in each blank
Show the answer and working
Answer: 16, 64, 64
4x squared is 16x2; twice the product, 2 × 4x × 8 = 64x; 8 squared is 64.
Example 3
Work it out quickly with a perfect square: 742 =
Show the answer and working
Answer: 5476
74 = 70 + 4: (70 + 4)2 = 4900 + 560 + 16 = 5476.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.(x − 9)2 = ?
- A. x2 + 81
- B. x2 − 18x + 81
- C. x2 − 9x + 81
- D. x2 − 81
Choose the right answer
Answer:
2.(3x − 6)2 ≡ px2 + qx + r, so p = , q = and r = .
Fill in each blank
Answer:
3.(−x − 5)2 = ?
- A. −x2 − 10x − 25
- B. x2 + 25
- C. x2 − 10x + 25
- D. x2 + 10x + 25
Choose the right answer
Answer:
4.(6x + 5y)2 = x2 + xy + y2
Fill in each blank
Answer:
5.Work it out quickly with a perfect square: 422 =
Answer:
All the answers
- 1. B. x2 − 18x + 81
- 2. 9, −36, 36
- 3. D. x2 + 10x + 25
- 4. 36, 60, 25
- 5. 1764
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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