+2653 The roots of the quadratic equation x^2 + bx + c are 5 + 3i and 1 - i. What is b + c?
+9673 Recall that
If the roots of the quadratic equation \(x^2 + bx + c = 0\) are \(\alpha\) and \(\beta\), then \(\alpha + \beta = -b\) and \(\alpha \beta = c\).
(Vieta's formula)
By this fact, we have \(b = -(5 + 3i) - (1 - i) = -6 - 2i\), \(c = (5 + 3i)(1 - i) = 5 - 2i - 3i^2 = 8 - 2i\).
Then, \(b + c = -6-2i + (8-2i) = \boxed{2 - 4i}\)
