Grade 9 Algebraic Expressions: Short Notes

9 ශ්‍රේණිය · 1 වාරය · 6 පරිච්ඡේදය · වීජීය ප්‍රකාශන

83%

of Grade 9 term tests include it (33 of 40)

272

marks in past papers (69 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Simplify expressions by collecting like terms
Find the value of an expression by substituting integers and fractions
Expand the product of two binomials
Expand (a + b)² and (a − b)² and use them
Form and simplify expressions for areas

The Avurudu sweet bill

A kade in Kegalle sells kavum at Rs x each and kokis at Rs y each. The bill for 5 kavum and 3 kokis is 5x + 3y rupees. On Avurudu morning, x = 40 and y = 25, so the bill is Rs 275. That one expression works for any prices.

Terms and like terms

In 3a + 2b − a + 4b there are four terms. The number in front is the coefficient; a alone means 1a.Like terms have exactly the same letters and powers: 3a and −a are like; 3a and 2b are not.Collect like terms, keeping each term's sign:3a − a + 2b + 4b = 2a + 6b

⚠️ Mixing up terms

O/L reports show three slips:• Adding unlike terms: 2a + 3b is not 5ab. It stays 2a + 3b.• Losing the sign: in 5x − 2y − 3x, the 3x is −3x, so the answer is 2x − 2y.• Confusing sum and product: y + y = 2y, but y × y = y².

Substituting negatives and fractions

Replace each letter with its value in brackets, then follow BODMAS.If p = −3:• 2p² = 2 × (−3)² = 2 × 9 = 18 (square first, then × 2)• 5 − p = 5 − (−3) = 8If q = ½: 6q = 6 × ½ = 3

✏️ Three values at once

Find the value of 3a − 4b + a² when a = −2 and b = ½.

13a = 3 × (−2) = −6
24b = 4 × ½ = 2, so −4b = −2
3a² = (−2)² = 4
4Total: −6 − 2 + 4 = −4

✅ −4

Multiplying two binomials

A binomial has two terms, like x + 5. To expand (x + 5)(x + 2), multiply each term of the first by each term of the second:x × x + x × 2 + 5 × x + 5 × 2= x² + 2x + 5x + 10= x² + 7x + 10Four products, then collect the like terms.

Why four products?

A rectangle with sides (x + 5) and (x + 2) splits into four parts. Their areas are the four products.

✏️ Watch the signs

Expand (i) (x + 5)(x − 3) (ii) (2x − 1)(x + 4)

1(i) x² − 3x + 5x − 15
2= x² + 2x − 15 (since +5 × −3 = −15)
3(ii) 2x² + 8x − x − 4
4= 2x² + 7x − 4

✅ (i) x² + 2x − 15 (ii) 2x² + 7x − 4

🧮 Square of a binomial

(a + b)² means (a + b)(a + b). The middle term 2ab comes from ab + ba. In (a − b)², only the middle term is negative.

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

⚠️ The missing middle term

Writing (x + 3)² = x² + 9 is the most common slip. Test it with x = 1: (1 + 3)² = 16, but 1 + 9 = 10.The right answer is x² + 6x + 9. Whenever you see a square bracket, write it twice, (x + 3)(x + 3), and expand.

A mental maths trick

Expansions make some squares easy without a calculator:101² = (100 + 1)² = 10 000 + 200 + 1 = 10 20149² = (50 − 1)² = 2500 − 100 + 1 = 2401Pick a round number close by and use the formula.

⚠️ Subtracting without brackets

O/L examiners report sign errors from subtracting without brackets. For the pond, writing x² + 8x + 12 − x² + 2x + 1 gives 10x + 13, which is wrong. The minus sign belongs to the whole pond area, so every term inside changes sign: − x² − 2x − 1.Always put the expression being subtracted in brackets first.

📌 Remember

• Collect only like terms, and keep each term's sign.• Substitute values in brackets; square before multiplying.• (x + a)(x + b): each term × each term, then collect.• (a ± b)² = a² ± 2ab + b²; never forget the middle term.• Put a subtracted expression in brackets before removing them.

🧾 Quick revision sheet

Like terms	Terms with exactly the same letters and powers, e.g. 3a² and −5a².
Coefficient of a in 7 − a	−1
(−4)² = ?	16 (a negative squared is positive)
(x + a)(x + b)	x² + (a + b)x + ab
(a + b)²	a² + 2ab + b²
(a − b)²	a² − 2ab + b²
Is (x + 3)² = x² + 9?	No: x² + 6x + 9.
y × y and y + y	y² and 2y