Question

Find the solutions of the quadratic equation 9x² + 12x + 8 = 0 in simplest a + bi form.

a) x = -1 - i, -1 + i
b) x = -2 - 2i, -2 + 2i
c) x = -1 - 3i, -1 + 3i
d) x = -4 - i, -4 + i

Answer 1

The discriminant of the quadratic equation 9x² + 12x + 8 = 0 is negative, indicating complex roots. Using the quadratic formula yields solutions -2/3 - 2/3i and -2/3 + 2/3i, which do not match with the given options, suggesting an error in the question or options.

Step-by-step explanation:

To find the solutions of the quadratic equation 9x² + 12x + 8 = 0, we can use the quadratic formula -b ± √b² - 4ac over 2a. In this equation, a = 9, b = 12, and c = 8.

The discriminant (√b² - 4ac) part of the formula will help us determine if the solutions are real or complex. Let's calculate the discriminant:

√(12)² - 4 × 9 × 8 = √144 - 288 = √-144

Since the discriminant is negative, the roots of the equation will be complex.

Substituting a, b, and c into the quadratic formula gives:

-12 ± √-144 over 18

Since √-144 is 12i (where i is the imaginary unit), the equation simplifies to:

-12 ± 12i over 18

Dividing by 18 gives us:

-2/3 ± 2/3i

Simplifying further, our solutions are:

x = -2/3 - 2/3i, x = -2/3 + 2/3i

However, this doesn't match with any of the given options. It seems there might be a mistake in the question or the provided options. The student should verify the original question for any errors or provide the correct options for a precise answer.