Divide (6-3i) by (2+i)
I would muliply numerator and denominator by 2-i. Then it becomes much easier.
To divide complex numbers, we use the technique of multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of the imaginary part.
In this case, the denominator is (2 + i), so its conjugate is (2 - i).
Now, let's multiply the numerator and denominator by the conjugate of the denominator:
(6 - 3i) * (2 - i) / (2 + i) * (2 - i)
Expanding the numerator and denominator:
((6 * 2) + (6 * (-i)) + (-3i * 2) + (-3i * (-i))) / ((2 * 2) + (2 * (-i)) + (i * 2) + (i * (-i)))
Simplifying the multiplication:
(12 - 6i - 6i + 3i^2) / (4 - 2i + 2i - i^2)
Note that i^2 = -1, so the expression becomes:
(12 - 12i + 3*(-1)) / (4 - (-1))
Simplifying further:
(12 - 12i - 3) / 4 + 1
Combining like terms:
(9 - 12i) / 5
The final result is 9/5 - (12/5)i.