which of the following expressions is equal to x^2+9? A. (x+3i)^2 B. (x+3i)(x-6i) C. (x-3i)^2 D. (x+3i)(x-3i)

Answer 1

To be sure that you get the correct answer, I would recommend that you multiply out all 4 expressions (A through D).

A quick way to get to the answer on this problem is to realize that x^2 + 9 does not have a middle "Axi" term. Therefore, the "Axi" terms will need to add up to zero. You will see that only occurs with option D because 3xi - 3xi = 0. Therefore, D is the correct answer.

Multiplying each out to show the full work -

A: (x + 3i) ^ 2 = (x + 3i) * (x + 3i) = x^2 + x3i + x3i + 9(i^2) = x^2 + 6xi - 9

B: (x + 3i) * (x - 6i) = x^2 + 3xi - 6xi - 18(i^2) = x^2 - 3xi + 18

C: (x - 3i) ^ 2 = (x - 3i) * (x - 3i) = x^2 - 3xi - 3xi + 9(i^2) = x^2 - 6xi + 9

D: (x + 3i) * (x - 3i) = x^2 + 3xi - 3xi - 9(i^2) = x^2 + 9

D is the correct answer.

Answer 2

Since, the unit imaginary number is i = √(-1); i² = -1;

The expression is:

x² + 9 = x² - (-9) = x² - 9*i²

Using the difference of two squares expansion:

a² - b² = (a - b)(a + b)

Then,

x² - 9*i² = x² - (3*i)² = (x - 3*i)(x + 3*i)

The fourth option D. would be the correct answer.