Find the value of z if z^2 + 4z + 8 = 0.

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Question: Find the value of z if z^2 + 4z + 8 = 0.

Options:

Correct Answer: -2 + 2i

Solution:

Using the quadratic formula, z = [-4 ± √(16 - 32)]/2 = -2 ± 2i.

Find the value of z if z^2 + 4z + 8 = 0.

Find the value of z if z^2 + 4z + 8 = 0.

Correct Answer: -2 ± 2i

Step 1: Identify the equation you need to solve, which is z^2 + 4z + 8 = 0.
Step 2: Recognize that this is a quadratic equation in the form of az^2 + bz + c = 0, where a = 1, b = 4, and c = 8.
Step 3: Use the quadratic formula, which is z = [-b ± √(b² - 4ac)] / (2a).
Step 4: Calculate b² - 4ac. Here, b = 4, a = 1, and c = 8. So, b² = 4² = 16 and 4ac = 4 * 1 * 8 = 32.
Step 5: Substitute these values into the formula: 16 - 32 = -16.
Step 6: Now, substitute b and the result into the quadratic formula: z = [-4 ± √(-16)] / (2 * 1).
Step 7: Calculate √(-16). Since the square root of a negative number involves 'i', we have √(-16) = 4i.
Step 8: Substitute this back into the formula: z = [-4 ± 4i] / 2.
Step 9: Simplify the expression: z = -2 ± 2i.

Quadratic Equations – Understanding how to solve quadratic equations using the quadratic formula.
Complex Numbers – Recognizing and working with complex numbers resulting from negative discriminants.