Two complex numbers are $u = 3 + 2i$ and $v = -1 + i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 3 - 2014

To plot the complex numbers on the Argand diagram:

• For $u = 3 + 2i$, plot the point (3, 2).
• For $v = -1 + i$, plot the point (-1, 1).
• For $w = 2 + i$, plot the point (2, 1).

Place these points in their respective locations on the diagram.

First, we simplify the top line:

$$2u + v = 2(3 + 2i) + (-1 + i) = 6 + 4i - 1 + i = 5 + 5i.$$

To divide these two complex numbers, we will multiply the fraction top and bottom by the conjugate of the denominator, $w$:

$$\frac{2u + v}{w} = \frac{5 + 5i}{2 + i}.$$

Now, multiplying both the numerator and the denominator by the conjugate of the denominator, $2 - i$:

$$\frac{(5 + 5i)(2 - i)}{(2 + i)(2 - i)} = \frac{10 - 5i + 10i - 5i^2}{4 + 1}.$$

Recall that $i^2 = -1$, so this simplifies to:

$$\frac{10 + 5 + 5i}{5} = \frac{15 + 5i}{5} = 3 + i.$$

Thus, we find:

$\frac{2u + v}{w} = 3 + i.$