1. **Problem statement:** Factorise each expression completely. 2. **Formula used:** Difference of squares: $$a^2 - b^2 = (a-b)(a+b)$$ 3. **(a) Factorise** $$(a + 3)^2 - 9$$ - Recognize $9 = 3^2$, so this is a difference of squares. - Apply formula: $$(a+3 - 3)(a+3 + 3) = (a)(a+6)$$ 4. **(b) Factorise** $$16 - 25(b + 3)^2$$ - Recognize $16 = 4^2$ and $25 = 5^2$. - Apply difference of squares: $$(4 - 5(b+3))(4 + 5(b+3))$$ - Simplify: $$(4 - 5b - 15)(4 + 5b + 15) = (-5b - 11)(5b + 19)$$ 5. **(c) Factorise** $$c^2 - (d + 2)^2$$ - Difference of squares: $$(c - (d+2))(c + (d+2)) = (c - d - 2)(c + d + 2)$$ 6. **(d) Factorise** $$(2h - 1)^2 - 4k^2$$ - Recognize $4k^2 = (2k)^2$. - Difference of squares: $$(2h - 1 - 2k)(2h - 1 + 2k)$$ 7. **(e) Factorise** $$(3x - 5)^2 - 169$$ - Recognize $169 = 13^2$. - Difference of squares: $$(3x - 5 - 13)(3x - 5 + 13) = (3x - 18)(3x + 8)$$ 8. **(f) Factorise** $$(p + 1)^2 - (p - 1)^2$$ - Use difference of squares: $$(p+1 - (p-1))(p+1 + (p-1))$$ - Simplify: $$(p+1 - p + 1)(p+1 + p - 1) = (2)(2p) = 4p$$ 9. **Evaluate** $$5 imes 88^2 - 720$$ - Calculate $88^2 = 7744$ - Multiply: $5 imes 7744 = 38720$ - Subtract: $38720 - 720 = 38000$ 10. **(i) Factorise** $$x^2 - 4y^2$$ - Recognize $4y^2 = (2y)^2$ - Difference of squares: $$(x - 2y)(x + 2y)$$ 11. **(ii) Solve** $$x^2 - 4y^2 = 13$$ for positive integers $x,y$ - Rewrite as $$(x - 2y)(x + 2y) = 13$$ - Since 13 is prime, possible factor pairs: $(1,13)$ or $(13,1)$ - Set $x - 2y = 1$, $x + 2y = 13$ - Add: $2x = 14 ightarrow x = 7$ - Subtract: $4y = 12 ightarrow y = 3$ - Check: $7^2 - 4 imes 3^2 = 49 - 36 = 13$ - Solution: $(x,y) = (7,3)$ **Final answers:** (a) $a(a+6)$ (b) $(-5b - 11)(5b + 19)$ (c) $(c - d - 2)(c + d + 2)$ (d) $(2h - 1 - 2k)(2h - 1 + 2k)$ (e) $(3x - 18)(3x + 8)$ (f) $4p$ Evaluation: $38000$ (i) $(x - 2y)(x + 2y)$ (ii) $(x,y) = (7,3)$