Final answer:
To find the coefficients b and c in the equation x² + bx + c = 0 that are also its roots, we use the relationships of roots and coefficients of a quadratic equation. Solving for b and c using these relationships reveals the coefficients to be b = 1 and c = -2.
Step-by-step explanation:
The student is inquiring about finding the coefficients b and c in a quadratic equation x² + bx + c = 0, given that the coefficients themselves are also the roots of the equation. To solve this problem, we use the fact that the sum and product of the roots of a quadratic equation ax² + bx + c = 0 are -b/a and c/a, respectively. Since in our case, a is 1, the sum of the roots b and c is -b and the product is c. Therefore, b and c must satisfy the equations b + c = -b and b*c = c.
From the first equation, we get 2b = -c, and using the second equation, we substitute to find c: (-c/2)*c = c, which simplifies to c²/2 = -c. We can solve for c to find that c must be equal to -2, since c is not 0 by the problem's restriction. Substituting back into 2b = -c, we find b is equal to 1. Thus, the coefficients in question are b = 1 and c = -2.