In a rectangle, the sum of adjacent sides is 41, and the diagonals are 23, find these sides.

Let the length of one of the adjacent sides of the rectangle be x centimeters, then the length of the second side is 41 x centimeters. Knowing that the diagonal is 23 centimeters (according to the Pythagorean theorem), we compose the equation:

x ^ 2 + (41 – x) ^ 2 = 23 ^ 2;

x ^ 2 + 41 ^ 2 – 82x + x ^ 2 = 529;

2x ^ 2 – 82x + 1681 – 529 = 0;

2x ^ 2 – 82x + 1152 = 0;

a = 2, b = -82, c = 1152;
D = b ^ 2 – 4 * a * c = 6724 – 4 * 2 * 1152 = 6724 – 9216 = -2492 (the discriminant is less than zero, then this quadratic equation does not have two roots);
Answer: there are no solutions.