Grade 10 Binomial Expressions: Short Notes

10 ශ්‍රේණිය · 1 වාරය · 4 පරිච්ඡේදය · ද්විපද ප්‍රකාශන

57%

of Grade 10 term tests include it (31 of 54)

245

marks in past papers (46 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Expand (a + b)² and (a − b)² quickly and correctly
Show the expansion of (a + b)² as areas of a square
Find the missing term that makes a perfect square
Square numbers like 102 and 49 in your head using a binomial
Find a² + b² from a + b (or a − b) and ab

Seeya's vegetable plot

Seeya's square vegetable plot in Kurunegala has sides of x metres. He widens every side by 3 m. Is the new area x² + 9? Try x = 10: the real area is 13² = 169 m², not 109 m². Squaring a binomial hides a middle part that most people forget.

What does squaring a binomial mean?

A binomial has two terms, like x + 3 or 2a − 5.Squaring it means multiplying it by itself:(x + 3)² = (x + 3)(x + 3)= x² + 3x + 3x + 9= x² + 6x + 9The two middle products are equal, so they join to make 6x.

(a + b)² as an area

A square of side a + b splits into a², b² and two rectangles of area ab. So (a + b)² = a² + 2ab + b².

🧮 Square of a sum

Say it as: first term squared, plus 2 × first × second, plus second term squared. Here a and b can be numbers, letters or terms like 3x.

(a + b)² = a² + 2ab + b²

✏️ Expanding squares of sums

Expand (i) (x + 4)² (ii) (2x + 3)²

1(i) First² = x², 2 × x × 4 = 8x, second² = 16.
2So (x + 4)² = x² + 8x + 16.
3(ii) First² = (2x)² = 4x², not 2x².
4Middle: 2 × 2x × 3 = 12x. Last: 3² = 9.
5So (2x + 3)² = 4x² + 12x + 9.

✅ (i) x² + 8x + 16 (ii) 4x² + 12x + 9

🧮 Square of a difference

Same pattern, but the middle term is negative. The last term is always positive, because (−b) × (−b) = +b².

(a − b)² = a² − 2ab + b²

✏️ Watch the signs

Expand (i) (3y − 5)² (ii) (5 − 2p)²

1(i) (3y)² = 9y²; 2 × 3y × 5 = 30y; 5² = 25.
2The middle term takes the minus sign: (3y − 5)² = 9y² − 30y + 25.
3(ii) Here a = 5 and b = 2p: 25 − 2 × 5 × 2p + 4p².
4So (5 − 2p)² = 25 − 20p + 4p².

✅ (i) 9y² − 30y + 25 (ii) 25 − 20p + 4p²

⚠️ (a + b)² is NOT a² + b²

In one O/L paper only 3 in 100 candidates got a binomial-square question right; most wrote (x + y)² = x² + y². Test it with numbers: (3 + 4)² = 7² = 49, but 3² + 4² = 25. The missing 24 is the 2ab = 2 × 3 × 4. Never drop the middle term.

Completing a perfect square

x² + 10x + ☐ will be a perfect square if ☐ is the square of half the x coefficient.Half of 10 is 5, and 5² = 25.So x² + 10x + 25 = (x + 5)².For x² − 10x + 25 it is (x − 5)²: the sign of the middle term goes into the bracket.

Square big numbers in your head

Split the number next to a round number.102² = (100 + 2)² = 10000 + 400 + 4 = 1040498² = (100 − 2)² = 10000 − 400 + 4 = 96049.9² = (10 − 0.1)² = 100 − 2 + 0.01 = 98.01

🧮 Finding a² + b² without a and b

Rearrange the expansions. Use the first form when you know a + b and ab, the second when you know a − b and ab.

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

⚠️ Forgetting to subtract 2ab

Examiners report students writing x² + y² = (x + y)² = 49, or not knowing how to start. (x + y)² also contains 2xy, so take it away: 49 − 20. Write the identity first, then substitute. Another slip: in a gap-fill like (☐ + 3)², put 3 in the bracket, not 9.

📌 Remember

• (a + b)² = a² + 2ab + b²• (a − b)² = a² − 2ab + b² (last term always +)• Never forget the middle term 2ab, and square the whole first term: (3x)² = 9x².• Perfect square: add (half the x coefficient)².• a² + b² = (a + b)² − 2ab = (a − b)² + 2ab

🧾 Quick revision sheet

(a + b)² = ?	a² + 2ab + b²
(a − b)² = ?	a² − 2ab + b²
What is a binomial?	An expression with two terms, e.g. 2x − 5.
Term that makes x² + 8x + ☐ a perfect square	16, because (8 ÷ 2)² = 16: (x + 4)²
a² + b² in terms of a + b and ab	(a + b)² − 2ab
a² + b² in terms of a − b and ab	(a − b)² + 2ab
103² using a binomial	(100 + 3)² = 10000 + 600 + 9 = 10609
If x + 1/x = k, then x² + 1/x² = ?	k² − 2