Moderate Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

In this moderate level, we'll explore factorisation by grouping, quadratic expressions, and the difference of squares.

These techniques are essential for solving equations and advanced algebraic manipulations.

1. What is the first step in factorisation by common factor method?

The first step is to find the HCFfind the LCM of all terms.

Perfect! Always find the highest common factor first.

2. What is the highest common factor (HCF) of 9x²y and 6xy²?

Answer:

Correct! HCF of 9 and 6 is 3, and of x²y and xy² is xy.

3. What type of algebraic expression can be factorised using grouping?

Expressions with four termsmultiple terms that can be grouped. (Note: Any similar answer)

Excellent! Usually expressions with 4 terms like ax + ay + bx + by.

4. Expand and simplify: x(a + b) + y(a + b).

= (a + b)( + )

Great! Common factor is (a + b), so the answer is (a + b)(x + y).

5. Factorise x² + 3x.

= (x + )

Perfect! x² + 3x = x(x + 3).

Drag each expression to its appropriate factorisation method:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Factorise by grouping: xy + xz + 3y + 3z.

Group: (xy + xz) + (3y + 3z)

= (y + z) + (y + z)

= (y + z)( + )

Excellent! xy + xz + 3y + 3z = (y + z)(x + 3).

2. Factorise: ax + ay + bx + by.

Group: (ax + ay) + (bx + by)

= (x + y) + (x + y)

= (x + y)( + )

Perfect! ax + ay + bx + by = (x + y)(a + b).

3. Factorise: 2x² + 7x + 3.

We need two numbers that multiply to 2 × 3 = and add to

Those numbers are and

= 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3)

= ( + )(x + )

Excellent! 2x² + 7x + 3 = (2x + 1)(x + 3).

4. Factorise: 6p² + 11p + 3.

Product = 6 × 3 = , Sum =

Numbers: and

= 6p² + 9p + 2p + 3 = 3p(2p + 3) + 1(2p + 3)

= ( + )( + )

Perfect! 6p² + 11p + 3 = (3p + 1)(2p + 3).

5. Factorise: x² + 6x + 8.

We need two numbers that multiply to and add to

Those numbers are and

= (x + )(x + )

Great! x² + 6x + 8 = (x + 4)(x + 2).

Part A: Section C – Long Answer Questions (4 Marks Each)

1. Factorise x² + 5x + 6 completely and verify your result by substitution.

We need two numbers that multiply to and add to

Those numbers are and

So, x² + 5x + 6 = (x + )(x + )

Verification: Let x = 1

LHS = 1² + 5(1) + 6 = 1 + 5 + 6 =

RHS = (1 + 3)(1 + 2) = 4 × 3 =

Verified! LHS = RHS = 12.

2. Factorise 3a²b + 6ab² + 9a³b² by taking common factors step-by-step.

Step 1: Find HCF of coefficients: HCF(3, 6, 9) =

Step 2: Find HCF of variables: HCF(a²b, ab², a³b²) =

Step 3: Overall HCF =

Step 4: Divide each term by 3ab:

3a²b ÷ 3ab =

6ab² ÷ 3ab =

9a³b² ÷ 3ab =

Final Answer: = (a + 2b + 3a²b)

Excellent! 3a²b + 6ab² + 9a³b² = 3ab(a + 2b + 3a²b).

3. Factorise x² - 49 using the difference of squares method.

This is in the form a² - b² = (a + b)(a - b)

Here, a = and b =

So, x² - 49 = (x + )(x - )

Perfect! x² - 49 = (x + 7)(x - 7).

4. Factorise 4x² - 12x + 9 and check your result using the identity (a - b)² = a² - 2ab + b².

This looks like (a - b)² = a² - 2ab + b²

Here, a² = 4x², so a =

And b² = 9, so b =

Check middle term: 2ab = 2 × 2x × 3 = ✓

So, 4x² - 12x + 9 = ( - )²

Verification: Expand (2x - 3)²

= (2x)² - 2(2x)(3) + 3² = 4x² - 12x + 9 ✓

Perfect! 4x² - 12x + 9 = (2x - 3)².

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. Factorisation of 2x² - 18 =

(a) 2(x² - 9) (b) 2(x - 3)(x + 3) (c) Both a and b (d) None

Correct! Both are valid: 2(x² - 9) = 2(x - 3)(x + 3). Complete factorisation is 2(x - 3)(x + 3).

7. The middle term in (x + 2)(x + 3) is:

(a) 5x (b) 6x (c) 10x (d) x²

Perfect! Expand: x² + 3x + 2x + 6 = x² + 5x + 6. Middle term is 5x.

8. a² + 2ab + b² =

(a) (a - b)² (b) (a + b)² (c) (a + 2b)² (d) (a + b)(a - b)

Excellent! This is the perfect square trinomial: a² + 2ab + b² = (a + b)².

9. 4x² - 9 =

(a) (2x + 3)(2x - 3) (b) (x + 3)(x - 3) (c) (4x + 9)(4x - 9) (d) None

Perfect! 4x² - 9 = (2x)² - 3² = (2x + 3)(2x - 3).

10. The HCF of 12x³y and 8x²y² is:

(a) 4x²y (b) 2x² (c) 8xy (d) 6xy²

Correct! HCF(12,8) = 4, HCF(x³,x²) = x², HCF(y,y²) = y. So HCF = 4x²y.

🎉 Outstanding Work! You've Mastered Intermediate Factorisation!

Here's what you learned:

• Factorisation by Grouping:

  • Used for 4-term expressions
  • Group terms with common factors
  • Factor out common binomial
  • Example: xy + 3y + 2x + 6 = (x + 2)(y + 3)

• Quadratic Factorisation (ax² + bx + c):

  • Find two numbers that multiply to ac and add to b
  • Split the middle term
  • Factor by grouping
  • Example: x² + 5x + 6 = (x + 2)(x + 3)

• Difference of Squares: a² - b² = (a + b)(a - b)

  • Examples: x² - 9 = (x + 3)(x - 3)
  • 4x² - 25 = (2x + 5)(2x - 5)

• Perfect Square Trinomials:

  • a² + 2ab + b² = (a + b)²
  • a² - 2ab + b² = (a - b)²
  • Example: x² - 6x + 9 = (x - 3)²

• Strategy:

  1. Look for common factors first
  2. Identify the type of expression
  3. Apply appropriate method
  4. Always verify by expanding

These techniques are crucial for solving quadratic equations and advanced algebraic problems!