Divide the complex number \$(2 - 3i)/(1 + 2i)\$ and express your answer in standard form.

Given that

\$(2 - 3i)/(1 + 2i)\$

Identify the conjugate of the denominator:

The denominator is 1+2i, and its conjugate is 1−2i.

Multiply the numerator and the denominator by the conjugate of the denominator:

\$((2 - 3i)/(1 + 2i)) \times ((1 - 2i)/(1 - 2i))\$

Expand the numerator: Apply the distributive property (FOIL method):

(2 - 3i)(1 - 2i)

\$2 \times 1 + 2(-2i) + (-3i)1 + (-3i) \times (-2i)\$

\$2 - 4i - 3i + 6i^2\$

Since \$i^2 = -1\$:

\$2 - 4i - 3i + 6(-1)\$

\$2 - 4i - 3i - 6\$

Combine like terms:

(2 - 6) + (-4i - 3i)

⇒ -4 - 7i

Expand the denominator:

(1 + 2i)(1 - 2i)

Apply the difference of squares formula (a + bi)(a - bi) = a^2 - (bi)^2 :

\$1^2 - (2i)^2\$

5

Combine the results:

\$(-4 - 7i)/5\$

Separate the real and imaginary parts:

\$(-4/5) + (-7i/5)\$

⇒ -0.8 - 1.4i

Therefore the division of \$(2 - 3i)/(1 + 2i)\$ is -0.8 - 1.4i.

This option represents the result we calculated. It shows that when you divide 2 - 3i by 1 + 2i, you get -0.8 - 1.4i, which is the correct answer

-0.8 - 1.4i

This option is incorrect. It has the wrong signs for both the real and imaginary parts. It suggests that the result is positive, but our calculation resulted in a negative value

0.8 + 1.4i

This option is incorrect. While the signs are correct compared to option B, the values are inaccurate. The real part should be negative and the imaginary part should be -1.4i, not positive

0.8 + 1.6i

This option is incorrect. It has the wrong signs and values. The real part should be negative and the imaginary part should be -1.4i, not positive

0.8 - 1.6i

Option

A

Can you summarize what you’ve understood in the above steps?