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.
