Final answer:
The roots of the quadratic equation x² + 18x + 90 = 0 are found using the quadratic formula, resulting in complex roots -9 + 3i and -9 - 3i.
Step-by-step explanation:
The roots of the quadratic equation x² + 18x + 90 = 0 can be found by applying the quadratic formula, which is used to solve equations of the form ax² + bx + c = 0. The quadratic formula states that for any quadratic equation ax² + bx + c = 0, the solutions for x can be found using:
x = [-b ± sqrt(b² - 4ac)] / (2a)
In this case, a = 1, b = 18, and c = 90. Plugging these values into the quadratic formula, we get:
x = [-18 ± sqrt((18)² - 4(1)(90))] / 2(1)
x = [-18 ± sqrt(324 - 360)] / 2
x = [-18 ± sqrt(-36)] / 2
Since the discriminant (b² - 4ac) is negative, we have complex roots. To find the complex roots, we must remember that sqrt(-1) is represented by i. Therefore:
x = (-18 ± 6i) / 2
x = -9 ± 3i
The roots of the equation in its simplest a+bi form are -9 + 3i and -9 - 3i.