Find two numbers if you know that their sum is 2, and the sum of the squares of these numbers is 100.

Let the first number be x and the second number be y. The sum of these numbers is (x + y) or 2. The sum of the squares of these numbers is (x ^ 2 + y ^ 2) or 100. Let’s compose a system of equations and solve it.

{x + y = 2; x ^ 2 + y ^ 2 = 100 – we express the variable x from the first equation of the system through y;

x = 2 – y – substitute the expression (2 – y) in the second equation instead of x;

(2 – y) ^ 2 + y ^ 2 = 100;

4 – 4y + y ^ 2 + y ^ 2 = 100;

2y ^ 2 – 4y + 4 – 100 = 0;

2y ^ 2 – 4y – 96 = 0;

y ^ 2 – 2y – 48 = 0;

D = b ^ 2 – 4ac;

D = (-2) ^ 2 – 4 * 1 * (-48) = 4 + 192 = 196; √D = 14;

x = (-b ± √D) / (2a);

y1 = (2 + 14) / 2 = 16/2 = 8 – the second number;

y2 = (2 – 14) / 2 = -12/2 = -6 – second number;

x1 = 2 – y1 = 2 – 8 = -6 – the first number;

x2 = 2 – y2 = 2 + 6 = 8.

The numbers can be 8 and (-6) or (-6) and 8, which are the same.

Answer. eight; -6.

Differently.

Let the first number be x, then the second number is (2 – x). By the condition of the problem, it is known that the sum of the squares of these numbers is equal to x ^ 2 + (2 – x) ^ 2 or 100. Let’s compose an equation and solve it.

x ^ 2 + (2 – x) ^ 2 = 100;

x ^ 2 + 4 – 4x + x ^ 2 – 100 = 0;

2x ^ 2 – 4x – 96 = 0;

x ^ 2 – 2x – 48 = 0;

x1 = 8; x2 = -6 are the first numbers;

2 – x1 = 2 – 8 = -6; 2 – x2 = 2 + 6 = 8 – these are the second numbers.

Answer. 8 and -6.