We can find the values of b and c using the relationship between the quadratic formula and the roots of the equation.

Roots and Quadratic Formula:

The quadratic formula relates the coefficients (a, b, and c) of the quadratic equation to its roots (r1 and r2):

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this case, we know the roots (5 + 3i and 5 - 3i) and want to find b + c.

Properties of Complex Roots:

Complex numbers as roots of quadratic equations always come in conjugate pairs. This means that if one root is a + bi (where i is the imaginary unit), the other root will be a - bi.

In this case, our roots are 5 + 3i and 5 - 3i, which confirms they are complex conjugates.

Sum of Roots and Coefficients:

There's a useful relationship between the roots (r1 and r2) of a quadratic equation and its coefficients (a, b, and c):

Sum of roots (r1 + r2) = -b / a

Product of roots (r1 * r2) = c / a

Since we're looking for b + c, and the quadratic has a leading coefficient of 1 (a = 1), we can use these relationships directly.

Finding b + c:

Sum of Roots:

The sum of the roots (5 + 3i) and (5 - 3i) is:

(5 + 5) + (3i - 3i) = 10

Since the sum of roots is also -b / a (and a = 1), we have:

-b = 10

Therefore, b = -10

Product of Roots:

The product of the roots (5 + 3i) and (5 - 3i) is:

(5 + 3i) * (5 - 3i) = 25 + 25 = 50

Since the product of roots is also c / a (and a = 1), we have:

c = 50

b + c:

Finally, add b and c to find the desired value:

b + c = -10 + 50 = 40

Therefore, b + c = 40.