The sum of two numbers is 25 and the product is 84. Find these numbers.

1. Let’s compose a system of two equations, denoting the unknown numbers x and y:

{x + y = 25; (one)
{xy = 84. (2)

2. By Vieta’s inverse theorem, the numbers x and y satisfying equations (1) and (2) are the roots of the reduced quadratic equation:

p ^ 2 – 25p + 84 = 0.

3. Let’s calculate the discriminant and solve the quadratic equation:

D = 25 ^ 2 – 4 * 84 = 625 – 336 = 289;

p = (25 ± √289) / 2 = (25 ± 17) / 2;

a) p1 = (25 – 17) / 2 = 8/2 = 4;
b) p2 = (25 + 17) / 2 = 42/2 = 21.
(x; y) = (4; 21); (21; 4).

Answer: (4; 21); (21; 4).