7) If (4 + 7i)(3 - 5i) + (2 + 3i) ^ 2 = a + ib find a and b
Answers
Answer:
Explanation:
4
+
5
i
2
−
3
i
Whenever we divide complex numbers we multiply both numerator and denominator with the complex conjugate of the denominator, this makes the denominator a real number
If the complex number is
a
+
i
b
then the complex conjugate is
a
−
i
b
For example
(
a
+
i
b
)
(
a
−
i
b
)
=
(
a
)
2
−
(
i
b
)
2
=
a
2
−
i
2
b
2
=
a
2
−
(
−
1
)
b
2
=
a
2
+
b
2
this is a real number.
Now back to our problem.
4
+
5
i
2
−
3
i
=
4
+
5
i
2
−
3
i
⋅
2
+
3
i
2
+
3
i
=
(
4
+
5
i
)
(
2
+
3
i
)
2
2
+
3
2
=
4
(
2
)
+
4
(
3
i
)
+
5
i
(
2
)
+
5
i
(
3
i
)
4
+
9
=
8
+
12
i
+
10
i
+
15
i
2
13
=
8
+
22
i
+
15
(
−
1
)
13
=
8
−
15
+
22
i
13
=
−
7
+
22
i
13
Answer
=
−
7
13
+
22
13
i
Answer in
a
+
i
b
form
Step-by-step explanation: