Question

Expressed in simplest a + bi form, (7 - 3i) + (x - 2i)^2 - (4i + 2x^2) is:

a) (3 - x^2) - (4x + 7)i
b) (3 + 3^2) - (4x + 7)i
c) (3 - x^2) - 7i
d) (3 + 3^2) - 7i

Answer 1

The expression (7 - 3i) + (x - 2i)^2 - (4i + 2x^2) simplifies to (3 - x^2) - (4x + 7)i, which is option a). This is accomplished by expanding the squared term, combining like terms, and grouping the real and imaginary parts separately.

Step-by-step explanation:

To simplify the expression (7 - 3i) + (x - 2i)^2 - (4i + 2x^2), first, we need to expand the square term and then combine like terms.

Step 1: Expand the square term

(x - 2i)^2 = x^2 - 4xi + 4i^2

Since i^2 = -1, the expression becomes x^2 - 4xi - 4.

Step 2: Now combine all the terms.

(7 - 3i) + (x^2 - 4xi - 4) - (4i + 2x^2)

Step 3: Simplify the expression by combining like terms.

7 + x^2 - 4 - 2x^2 - 3i - 4xi - 4i

Combine real parts and imaginary parts separately:

Real part: 7 + x^2 - 4 - 2x^2 = 3 - x^2

Imaginary part: -3i - 4xi - 4i = -7i - 4xi

Step 4: Write in a + bi form

(3 - x^2) - (4x + 7)i which corresponds to option a).