Sum of the areas of two squares is 400cm². If the difference of their perimeters is 16 𝑐𝑚, find the length of their sides.

We are given the sum of the areas of two squares and the difference between their perimeters, and we have to find the length of the sides of the squares.
Let the length of one square is 𝑥, and that of the other square be 𝑦. 

Now, the sum of the areas of two squares = 400 cm²
⇒ 𝑥² + 𝑦²   = 400                                             ...(1)
And the difference between their perimeters = 16 𝑐𝑚
⇒ 4𝑥 − 4𝑦 = 16
⇒ 4(𝑥 − 𝑦) = 16
⇒ 𝑥 − 𝑦 = 16/4
⇒ 𝑥 − 𝑦 = 4
⇒ 𝑥 = 4 + 𝑦                            ...... (2)
Putting the value of 𝑥 in eq(1), we get
(4 + 𝑦)² + 𝑦²   = 400
⇒ 16 + 𝑦² + 8𝑦 + 𝑦²   = 400
⇒ 2𝑦² + 8𝑦 + 16 = 400
⇒ 2𝑦² + 8𝑦 + 16 − 400 = 0
⇒ 2𝑦² + 8𝑦 − 384 = 0
⇒ 2(𝑦² + 4𝑦 − 192) = 0
⇒ 𝑦² + 4𝑦 − 192 = 0
⇒ 𝑦² + 16𝑦 − 12𝑦 − 192 = 0
⇒ 𝑦(𝑦 + 16) − 12(𝑦 + 16) = 0
⇒ (𝑦 − 12)(𝑦 + 16) = 0
⇒ 𝑦 = 12 and 𝑦 =− 16

As 𝑦 is the length of the side of the square, and it cannot be negative, the value of 𝑦 is
𝑦 = 12 𝑐𝑚.
Putting the value of 𝑦 in eq(2), we get
𝑥 = 4 + 12
⇒ 𝑥 = 16 𝑐𝑚
Hence, the sides of the squares are 16 𝑐𝑚 and 12 𝑐𝑚.