Find the sides of the rectangle if their difference is 23 dm and the diagonal of the rectangle is 37 dm.

Let one side of the rectangle be x dm and the other side of the rectangle be equal to y dm. It is known that the difference between the sides of a rectangle is equal to (x – y) dm or 23 dm. The sides of a rectangle and its diagonal form a right-angled triangle. In this right-angled triangle, the diagonal will be the hypotenuse, and the sides will be the legs. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs, i.e. x ^ 2 + y ^ 2 = 37 ^ 2. Let’s compose a system of equations and solve it.

{x – y = 23; x ^ 2 + y ^ 2 = 37 ^ 2 – from the first equation we express x through y;

x = 23 + y – substitute the expression (23 – y) in the second equation instead of x;

(23 + y) ^ 2 + y ^ 2 = 37 ^ 2;

529 + 46y + y ^ 2 + y ^ 2 = 1369;

2y ^ 2 + 46y + 529 – 1369 = 0;

2y ^ 2 + 46y – 840 = 0;

y ^ 2 + 23y – 420 = 0;

D = b ^ 2 – 4ac;

D = (23) ^ 2 – 4 * 1 * (-420) = 529 + 1680 = 2209; √D = 47;

x = (-b ± √D) / (2a);

y1 = (-23 + 47) / 2 = 24/2 = 12 (dm) – the first side of the rectangle;

y2 = (-23 – 47) / 2 = -35 – the length cannot be negative;

x = 23 + y = 12 + 23 = 35 (dm) – the second side.

Answer. 12 dm, 35 dm.