Simplify each of the following, giving your answer in the form a+bi, where a, b, e, z:
A). (3-4i)-(-6+5i)
B). (2-3i)^2/1+2i
A). (3-4i)-(-6+5i)
To simplify this, we can distribute the negative sign when subtracting:
(3-4i)+6-5i
Combine like terms:
9 - 9i
So the simplified form is 9 - 9i.
B). (2-3i)^2/1+2i
To simplify this, we can start by squaring the numerator:
(2-3i)(2-3i)
Using FOIL method:
4 - 6i - 6i + 9i^2
Simplify the middle terms and replace i^2 with -1:
4 - 12i + 9(-1)
Combine like terms:
4 - 12i - 9
-5 - 12i
Now let's simplify the denominator:
1+2i
To rationalize this, we multiply the numerator and denominator by the conjugate of 1+2i, which is 1-2i:
(-5-12i)(1-2i)/(1+2i)(1-2i)
Using FOIL method:
(-5+10i-12i+24i^2)/(1-2i+2i-4i^2)
Simplify the middle terms and replace i^2 with -1:
(-5-2i+24(-1))/(1-4(-1))
Simplify further:
(-5-2i-24)/(1+4)
Combine like terms:
(-29-2i)/5
So the simplified form is (-29-2i)/5.