1. **Problem Statement:** (a) Factorize completely: (i) $2x^3 - 8x$ (ii) $3x^2 - 5x - 2$ (b) (i) Make $C$ the subject of the formula $F = \frac{9}{5} C + 32$. (ii) Given $F = 113$, calculate $C$. (c) 500 tickets sold: $x$ tickets at 6 each, remainder at 10 each. (i) Write expressions for: a) Number of tickets sold at 10 each b) Total money collected (ii) Given total money collected is 4108, find number of tickets sold at 6 each. --- 2. **Step-by-step solution:** **(a)(i) Factorize $2x^3 - 8x$:** - Factor out the common factor $2x$: $$2x^3 - 8x = 2x(x^2 - 4)$$ - Recognize $x^2 - 4$ as a difference of squares: $$x^2 - 4 = (x - 2)(x + 2)$$ - So the complete factorization is: $$2x(x - 2)(x + 2)$$ **(a)(ii) Factorize $3x^2 - 5x - 2$:** - Find two numbers that multiply to $3 \times (-2) = -6$ and add to $-5$. - These numbers are $-6$ and $1$. - Rewrite middle term: $$3x^2 - 6x + x - 2$$ - Factor by grouping: $$3x(x - 2) + 1(x - 2)$$ - Factor out common binomial: $$(3x + 1)(x - 2)$$ **(b)(i) Make $C$ the subject of $F = \frac{9}{5} C + 32$:** - Subtract 32 from both sides: $$F - 32 = \frac{9}{5} C$$ - Multiply both sides by $\frac{5}{9}$: $$C = \frac{5}{9} (F - 32)$$ **(b)(ii) Calculate $C$ when $F = 113$:** - Substitute $F = 113$: $$C = \frac{5}{9} (113 - 32) = \frac{5}{9} \times 81 = 5 \times 9 = 45$$ **(c)(i)(a) Number of tickets sold at 10 each:** - Total tickets = 500 - Tickets sold at 6 each = $x$ - Tickets sold at 10 each = $500 - x$ **(c)(i)(b) Total money collected:** - Money from $x$ tickets at 6 each: $6x$ - Money from $500 - x$ tickets at 10 each: $10(500 - x)$ - Total money collected: $$6x + 10(500 - x)$$ **(c)(ii) Given total money collected is 4108, find $x$:** - Set up equation: $$6x + 10(500 - x) = 4108$$ - Expand: $$6x + 5000 - 10x = 4108$$ - Combine like terms: $$-4x + 5000 = 4108$$ - Subtract 5000: $$-4x = 4108 - 5000 = -892$$ - Divide both sides by -4: $$x = \frac{-892}{-4} = 223$$ **Final answers:** - (a)(i) $2x(x - 2)(x + 2)$ - (a)(ii) $(3x + 1)(x - 2)$ - (b)(i) $C = \frac{5}{9} (F - 32)$ - (b)(ii) $C = 45$ - (c)(i)(a) $500 - x$ - (c)(i)(b) $6x + 10(500 - x)$ - (c)(ii) $x = 223$