If the diagonal and the area of a rectangle are 25 m² and 168 m², what is the length of the rectangle ?

Solution(By Examveda Team)

Let the length of the rectangle be x metres.
Then, breath of the rectangle = $$\left( {\frac{{168}}{x}} \right)m$$
$$\eqalign{ & \therefore \sqrt {{x^2} + {{\left( {\frac{{168}}{x}} \right)}^2}} = 25 \cr & \Rightarrow \sqrt {{x^2} + \frac{{28224}}{{{x^2}}}} = 25 \cr & \Rightarrow {x^2} + \frac{{28224}}{{{x^2}}} = 625 \cr & \Rightarrow {x^4} - 625{x^2} + 28224 = 0 \cr & \Rightarrow {x^4} - 576{x^2} - 49{x^2} + 28224 = 0 \cr & \Rightarrow {x^2}\left( {{x^2} - 576} \right) - 49\left( {{x^2} - 576} \right) = 0 \cr & \Rightarrow \left( {{x^2} - 576} \right)\left( {{x^2} - 49} \right) = 0 \cr & \Rightarrow {x^2} = 576\,\,or\,\,{x^2} = 49 \cr & \Rightarrow x = 24\,\,\,or\,\,\,x = 7 \cr} $$
Hence, length = 24 m and breadth = 7 m