Final answer:
The roots of the equation x² + 16x + 65 = 0 are complex and can be found using the quadratic formula. The roots are -8 + i and -8 - i.
Step-by-step explanation:
The roots of the quadratic equation x² + 16x + 65 = 0 can be found using the quadratic formula, which is:
x = ∛(b² - 4ac)/(2a)
In this equation, a = 1, b = 16, and c = 65. Substituting these values into the quadratic formula we get:
x = [-16 ± √(16² - 4(1)(65))]/(2(1))
x = [-16 ± √(256 - 260)]/2
x = [-16 ± √(-4)]/2
Since the discriminant (-4) is negative, the roots are complex and can be written in a + bi form:
x = (-16/2) ± (√4/2)i
x = -8 ± 1i
Therefore, the roots are -8 + i and -8 - i.