Find the lower of two numbers that add 17 and sum their squares 185.

Let the first number be x, then the second number is (17 – x). The square of the first number is x ^ 2, and the square of the second number is (17 – x) ^ 2. By the condition of the problem, it is known that the sum of the squares of these two numbers is equal to (x ^ 2 + (17 – x) ^ 2) or 185. Let’s compose an equation and solve it.

x ^ 2 + (17 – x) ^ 2 = 185;

x ^ 2 + 289 – 34x + x ^ 2 = 185;

2x ^ 2 – 34x + 289 – 185 = 0;

2x ^ 2 – 34x + 104 = 0;

x ^ 2 – 17x + 52 = 0;

D = b ^ 2 – 4ac;

D = (-17) ^ 2 – 4 * 1 * 52 = 289 – 208 = 81; √D = 9;

x = (-b ± √D) / (2a);

x1 = (17 + 9) / 2 = 26/2 = 13 – the first number;

x2 = (17 – 9) / 2 = 8/2 = 4 – the first number;

17 – x1 = 17 – 13 = 4 – the second number;

17 – x2 = 17 – 4 = 13 is the second number.

Of the numbers 13 and 4 or 4 and 13, the smallest number is 4.

Answer. 4.