The problem has two parts. (a) Expand and simplify the expression $(n+4)^2 + (n-2)^2$. (b) At an electrical shop, out of 92 people, 45 bought cables and 58 bought sockets, while 14 just asked for prices without buying anything. (i) Represent the information on a Venn diagram. (ii) Calculate the number of people who bought both sockets and cables.

1. Problem Description

The problem has two parts.

(a) Expand and simplify the expression $(n+4)^2 + (n-2)^2$.

(b) At an electrical shop, out of 92 people, 45 bought cables and 58 bought sockets, while 14 just asked for prices without buying anything.

(i) Represent the information on a Venn diagram.

(ii) Calculate the number of people who bought both sockets and cables.

2. Solution Steps

(a) Expanding and simplifying $(n+4)^2 + (n-2)^2$.

$(n+4)^2 = n^2 + 2(n)(4) + 4^2 = n^2 + 8n + 16$

$(n-2)^2 = n^2 - 2(n)(2) + (-2)^2 = n^2 - 4n + 4$

$(n+4)^2 + (n-2)^2 = (n^2 + 8n + 16) + (n^2 - 4n + 4) = 2n^2 + 4n + 20$

(b) Let $C$ be the set of people who bought cables and $S$ be the set of people who bought sockets.

Total number of people $= 92$

Number of people who bought cables, $n(C) = 45$

Number of people who bought sockets, $n(S) = 58$

Number of people who bought nothing $= 14$

Number of people who bought either cables or sockets or both $= 92 - 14 = 78$

$n(C \cup S) = n(C) + n(S) - n(C \cap S)$

$78 = 45 + 58 - n(C \cap S)$

$78 = 103 - n(C \cap S)$

$n(C \cap S) = 103 - 78 = 25$

Therefore, the number of people who bought both sockets and cables is

2

5. Venn Diagram:

Draw two overlapping circles. Label one circle "Cables (C)" and the other "Sockets (S)".

The overlapping region represents people who bought both.

$n(C \cap S) = 25$

Only cables: $n(C) - n(C \cap S) = 45 - 25 = 20$

Only sockets: $n(S) - n(C \cap S) = 58 - 25 = 33$

Neither: $14$

3. Final Answer

(a) $2n^2 + 4n + 20$

(b)(ii) The number of people who bought both sockets and cables is

2

5.