Question

\frac{(2+5i)}{(3-2i)}

a. \frac{4}{13} + \frac{19}{13}i

b. -\frac{4}{13} - \frac{19}{13}i

c. -\frac{4}{13} + \frac{19}{13}i

d. \frac{4}{13} - \frac{19}{13}i

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of \(3 - 2i\) is \(3 + 2i\). So we multiply the fraction \(\frac{2 + 5i}{3 - 2i}\) by \(\frac{3 + 2i}{3 + 2i}\).

$$ \frac{(2 + 5i)(3 + 2i)}{(3 - 2i)(3 + 2i)} $$

Step2: Expand the numerator and the denominator.

First, expand the numerator using the distributive property (FOIL method):

$$
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$$

Since \(i^2=-1\), we substitute that in:

$$ 6 + 19i + 10\times(-1)=6 + 19i - 10=-4 + 19i $$

Now, expand the denominator using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 3\) and \(b = 2i\):

$$ 3^2-(2i)^2 = 9 - 4i^2 $$

Substitute \(i^2=-1\):

$$ 9 - 4\times(-1)=9 + 4 = 13 $$

Step3: Write the fraction in standard form.

Now we have \(\frac{-4 + 19i}{13}\), which can be split into the real and imaginary parts:

$$ \frac{-4}{13}+\frac{19}{13}i $$

Wait, no, wait. Wait, in step 2, when we expanded the numerator: \(2\times3 = 6\), \(2\times2i = 4i\), \(5i\times3 = 15i\), \(5i\times2i = 10i^2\). So \(6 + 4i + 15i + 10i^2 = 6 + 19i - 10 = -4 + 19i\)? Wait, no, 6 - 10 is -4, yes. Then denominator is 13. So \(\frac{-4 + 19i}{13}=-\frac{4}{13}+\frac{19}{13}i\)? But wait, let's check again. Wait, maybe I made a mistake in the numerator expansion. Wait, \( (2 + 5i)(3 + 2i) \): 23=6, 22i=4i, 5i3=15i, 5i2i=10i². So 6 + 4i + 15i + 10i². i² is -1, so 10*(-1)= -10. So 6 - 10 is -4, and 4i + 15i is 19i. So numerator is -4 + 19i. Denominator is 13. So \(\frac{-4 + 19i}{13}=-\frac{4}{13}+\frac{19}{13}i\), which is option C? Wait, but let's check the original problem again. Wait, the original fraction is \(\frac{2 + 5i}{3 - 2i}\). Wait, maybe I messed up the sign in the conjugate? Wait, no, the conjugate of \(a - bi\) is \(a + bi\), so that's correct. Wait, but let's recalculate the numerator:

Wait, \( (2 + 5i)(3 + 2i) = 23 + 22i + 5i3 + 5i2i = 6 + 4i + 15i + 10i² = 6 + 19i - 10 = -4 + 19i\). Denominator: \(3² - (2i)² = 9 - 4i² = 9 + 4 = 13\). So \(\frac{-4 + 19i}{13}=-\frac{4}{13}+\frac{19}{13}i\), which is option C. Wait, but let's check the options again. Option C is \(-\frac{4}{13}+\frac{19}{13}i\), yes. So that's the correct answer.

Wait, but wait, maybe I made a mistake in the numerator. Let's do it again:

\( (2 + 5i)(3 + 2i) = 23 + 22i + 5i3 + 5i2i = 6 + 4i + 15i + 10i² \). \(i² = -1\), so 10i² = -10. So 6 - 10 = -4, 4i + 15i = 19i. So numerator is -4 + 19i. Denominator is 13. So \(\frac{-4 + 19i}{13}=-\frac{4}{13}+\frac{19}{13}i\), which is option C. So the correct answer is C.

Answer:

C. \(-\frac{4}{13}+\frac{19}{13}i\)