+16 The roots of the quadratic equation \( z^2 + bz + c = 0\) are 5 + 3i and 5-3i. What is b+c?
+2653 Since the quadratic equation has real coefficients and the roots are complex numbers, the roots come in complex conjugate pairs. This means that the other root must be 5−3i.
We can use the fact that the sum of the roots of a quadratic equation is equal to the negative of the coefficient of our z term. In other words, the sum of the roots is b. So, we have:
$$ b = 5 + 3i + (5 - 3i) = 10. $$
We can also use the fact that the product of the roots of a quadratic equation is equal to the constant term. In other words, the product of the roots is c. So, we have:
$$ c = (5 + 3i)(5 - 3i) = 5^2 - (3i)^2 = 25 + 9 = 34. $$
Therefore, b + c = 10 + 34 = 44.
