Simplify each of the following giving your answer in the form a+bi, where a,b,e,z:
(2-3i)^2/1+2i
To simplify (2-3i)^2/1+2i, let's first simplify the numerator:
(2-3i)^2 = (2-3i)(2-3i)
= 4 - 6i - 6i - 9i^2
= 4 - 12i + (-9)
= -5 - 12i
Next, let's simplify the denominator:
1+2i
To divide by a complex number, we need to rationalize the denominator. Multiply the denominator by its complex conjugate:
(1+2i)(1-2i)
= 1 - 2i + 2i -4i^2
= 1 - 4i^2
= 1 - 4(-1)
= 1 + 4
= 5
Now, we can simplify the expression:
(-5 - 12i) / 5
Divide each term by 5:
-5/5 - 12i/5
= -1 - (12/5)i
= -1 - 2.4i
Therefore, (2-3i)^2/1+2i simplifies to -1 - 2.4i.