The product of two consecutive natural numbers is greater than their sum by 109. Find these numbers?

Natural numbers are numbers used in counting objects: 1, 2, 3, …. Consecutive numbers differ from each other by 1.

Let the first number be x, then the second number is (x + 1). The product of these numbers is x (x + 1), and their sum is (x + (x + 1)) = x + x + 1 = 2x + 1. By the problem statement, it is known that the product of these numbers is greater than their sum by x ( x + 1) – (2x + 1) or 109. Let’s make an equation and solve it.

x (x + 1) – (2x + 1) = 109;

x ^ 2 + x – 2x – 1 = 109;

x ^ 2 – x – 1 – 109 = 0;

x ^ 2 – x – 110 = 0;

D = b ^ 2 – 4ac;

D = (-1) ^ 2 – 4 * 1 * (-110) = 1 + 440 = 441; √D = 21;

x = (-b ± √D) / (2a);

x1 = (1 + 21) / 2 = 22/2 = 11 – the first number;

x2 = (1 – 21) / 2 = -10 – not natural;

x + 1 = 11 + 1 = 12.

Answer. eleven; 12.