The area of a big rectangular room is 300 m². If the length were decreased by 5 m and breadth increased by 5 m; the area would be unaltered. Find the length of room.

Let length be x and breadth be y m.

xy = 300

y = $\dfrac{300}{x}$ …..(i)

According to question

$\Rightarrow (x - 5)(y + 5) = 300 \\[1em] \Rightarrow (x - 5)\Big(\dfrac{300}{x} + 5\Big) = 300 \\[1em] \Rightarrow 300 + 5x - \dfrac{1500}{x} - 25 = 300 \\[1em] \Rightarrow 5x - \dfrac{1500}{x} - 25 = 0 \\[1em] \Rightarrow 5x^2 - 25x - 1500 = 0 \\[1em] \Rightarrow x^2 - 5x - 300 = 0 \\[1em] \Rightarrow x^2 - 20x + 15x - 300 = 0 \\[1em] \Rightarrow x(x - 20) + 15(x - 20) = 0 \\[1em] \Rightarrow (x + 15)(x - 20) = 0 \\[1em] \Rightarrow x = -15 \text{ or } x = 20.$

Side cannot be negative,

∴ x = 20.

Hence, length of room is 20 m.