The product of two natural numbers is 154, and the sum of their square is 317. Find these numbers.

To solve the problem, you can compose a system of equations, and as it is known that the product of two numbers is 154, and the sum of their squares is 317. Let’s denote the numbers x and y:
x * y = 154
x² + y² = 317;
x = 154 / y
y⁴ – 317y² + 23716 = 0.
Let’s set z = y² and get the usual quadratic equation:
z² – 317z + 23716 = 0;
z₁ = 196; z₂ = 121;
y₁ = 14; y₂ = 11.
x₁ = 11; x₂ = 14.
Answer: x and y have two solutions: y₁ = 14; y₂ = 11; x₁ = 11; x₂ = 14.