Question

What are the roots of the equation x² + 12x + 40 = 0 in simplest a + bi form?

a) -6 + 2i, -6 - 2i
b) -2 + 6i, -2 - 6i
c) -10 + 4i, -10 - 4i
d) -4 + 10i, -4 - 10i

Answer 1

The roots of the given quadratic equation x² + 12x + 40 = 0 are −6 + i and −6 − i, which can be found using the quadratic formula.

Step-by-step explanation:

The student has presented a quadratic equation and is asked to find its roots in a + bi form, which requires using the quadratic formula. Given the equation x² + 12x + 40 = 0, we can find its roots using the quadratic formula:

x = −b ± √(b² − 4ac) / (2a)

Here, a = 1, b = 12, and c = 40. Substituting these values into the formula, we get:

x = −(12) ± √((12)² − 4(1)(40)) / (2(1))

x = −(12) ± √(144 − 160) / 2

x = −(12) ± √(−4) / 2

Since we have a negative value under the square root, we will have two complex roots. We can rewrite √(−4) as 2i, since √(−4) = √4 √(−1) = 2i.

So, the roots are:

x = (−(12) + 2i) / 2 and x = (−(12) − 2i) / 2

x = −6 + i and x = −6 − i

Thus, the roots of the equation in a + bi form are −6 + i and −6 − i.