If twice the area of a smaller square is subtracted from the area of a larger square, the result is 14cm². However, if twice the area of the larger square is added to three times the area of the smaller square, the result is 203 cm². Determine the sides of two squares.

Let the sides of the smaller and bigger square be x cm and y cm, respectively.

Area of smaller square = x² cm²

Area of larger square = y² cm²

According to question,

$y^2 - 2x^2 = 14 \\[1em] \Rightarrow y^2 = 14 + 2x^2 \qquad \text{[…Eq 1]}$

and

$3x^2 + 2y^2 = 203 \qquad \text{[…Eq 2]}$

Putting value of y² from Equation 1 in Equation 2, we get:

$\Rightarrow 3x^2 + 2(14 + 2x^2) = 203 \\[1em] \Rightarrow 3x^2 + 28 + 4x^2 = 203 \\[1em] \Rightarrow 7x^2 = 203 - 28 \\[1em] \Rightarrow 7x^2 = 175 \\[1em] \Rightarrow x^2 = 25 \\[1em] \Rightarrow x = \sqrt{25} \\[1em] x = 5 \text{ or } x = -5$

Since, side of a square cannot be negative hence, x ≠ -5.

∴ x = 5
y² = 14 + 2² = 64
⇒ y = 8

Hence, the length of larger square's side is 8cm and smaller square's is 5cm.