Given equation: x^2 + bx + c = 0

Equation after decreasing each root by one: (x - 1)^2 + b(x - 1) + c = 0

Equation given: x^2 + 8x + 2 = 0

Comparing the given equation with the decreased equation:
- The coefficient of x^2 is the same in both equations, so a = 1.
- The coefficient of x is 8 in the given equation, while in the decreased equation it is b - 1. So, b - 1 = 8, which implies b = 9.
- The constant term in the given equation is 2, while in the decreased equation it is c - b + 1. So, c - 9 + 1 = 2, which implies c = 10.

Therefore, the values of a, b, and c are:
- a = 1
- b = 9
- c = 10

Explanation:
To solve this problem, we first need to understand the given equation and the equation after decreasing each root by one.

The given equation is a quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.

When we decrease each root of the equation by one, we have to modify the equation accordingly. We subtract 1 from each root, which means we subtract 1 from each term that contains x. This gives us the equation (x - 1)^2 + b(x - 1) + c = 0.

To find the values of a, b, and c, we need to compare the given equation with the decreased equation.

By comparing the coefficients of the corresponding terms in both equations, we can determine the values of a, b, and c.

In this case, the coefficient of x^2 is the same in both equations, so we have a = 1.

The coefficient of x in the given equation is 8, while in the decreased equation it is b - 1. By equating these two, we find b - 1 = 8, which gives us b = 9.

Similarly, by comparing the constant terms in both equations, we can find the value of c. The constant term in the given equation is 2, while in the decreased equation it is c - b + 1. Equating these two, we get c - 9 + 1 = 2, which implies c = 10.

Therefore, the values of a, b, and c are 1, 9, and 10 respectively.