Let's solve the questions provided in the image step by step:

(a) Factorise

(i) $5a^2b + ab^2$

• Factor out the common factor $ab$: $5a^2b + ab^2 = ab(5a + b)$

(ii) $9k^2 - 1$

• This is a difference of squares: $9k^2 - 1 = (3k + 1)(3k - 1)$

(iii) $2y^2 - 5y + 2$

• Factorise using splitting the middle term: $2y^2 - 5y + 2 = 2y^2 - 4y - y + 2$ Group the terms: $= (2y^2 - 4y) - (y - 2)$ Factorise each group: $= 2y(y - 2) - 1(y - 2)$ Factorise completely: $= (2y - 1)(y - 2)$

(b) Expand and simplify

Expand $(2x + 5)(3x - 4)$: $(2x + 5)(3x - 4) = 2x \cdot 3x + 2x \cdot (-4) + 5 \cdot 3x + 5 \cdot (-4)$ Simplify: $= 6x^2 - 8x + 15x - 20$ Combine like terms: $= 6x^2 + 7x - 20$

(c) Adam, Imran, and Shakeel were playing a card game.

(i) Write down, in terms of $x$, an expression for the number of points scored by Shakeel.

• Adam scored $x$ points.
• Imran scored 3 points fewer than Adam: $x - 3$.
• Shakeel scored twice as many points as Imran: $\text{Shakeel's score} = 2(x - 3)$

(ii) Write an equation which may be used to find the value of $x$.

• Together, they scored 39 points: $\text{Adam's score} + \text{Imran's score} + \text{Shakeel's score} = 39$ Substitute their scores: $x + (x - 3) + 2(x - 3) = 39$ Simplify: $x + x - 3 + 2x - 6 = 39$ Combine like terms: $4x - 9 = 39$

Let me know if you'd like me to continue solving $4x - 9 = 39$ or need more details on any of the steps!

Questions to Expand Understanding:

1. What are the steps to factorise a quadratic expression like $2y^2 - 5y + 2$?
2. How do you identify a difference of squares and factorise it?
3. What are some common techniques for simplifying expanded algebraic expressions?
4. Can you write another real-life example of forming equations based on a scenario like (c)?
5. How would the solution for $4x - 9 = 39$ proceed to find $x$?

Tip:

Always double-check your factorisation by expanding it back to verify if it matches the original expression.