20 The diagram shows a rectangular playground of width \( x \) metres and length \( 3 x \) metres. The playground is extended, by adding 10 metres to its width and 20 metres to its length, to form a larger rectangular playground. The area of the larger rectangular playground is double the area of the original playground. (a) Show that \( 3 x^{2}-50 x-200=0 \) accurately drawn

Let's dive into the details of this problem! Initially, the original playground has an area of \( A = \text{width} \times \text{length} = x \times 3x = 3x^2 \) square metres. When we expand it, the new dimensions become \( x + 10 \) for width and \( 3x + 20 \) for length. Thus, the area of the larger playground is given by \( A' = (x + 10)(3x + 20) \). According to the problem, this area should equal double the area of the original playground, leading us to the equation \( (x + 10)(3x + 20) = 2(3x^2) \). Expanding the left-hand side, we have \( 3x^2 + 20x + 30x + 200 = 3x^2 + 50x + 200 \). Setting this equal to the right-hand side gives us \( 3x^2 + 50x + 200 = 6x^2 \). Rearranging leads to \( 3x^2 - 50x - 200 = 0 \), which is the required equation. Now we see how to reach that conclusion! To make your understanding of quadratic equations fun, picture this scenario: You’re on a treasure hunt in a pixelated world where every rectangle you explore expands the map. The smaller playground is like your starter zone, where you collect coins (areas) before leveling up! A larger playground means more treasures, and who doesn’t want double their loot? Just like leveling in a game, playing around with dimensions helps you unlock paths to solving those quadratic puzzles! If you want to delve deeper into the wonders of quadratic equations, many classic problem-solving resources unveil strategies for tackling them, such as factoring, completing the square, or the quadratic formula. Stories within math textbooks often showcase creative plots that integrate problem-solving scenarios. You might find inspiration in real-life applications of quadratics, such as determining the best fit for a garden layout or optimizing a park's area, where numbers dance to create perfect designs!