Final answer:
The quotient 9/6+3i in the form a+bi is found by multiplying the numerator and denominator by the complex conjugate of the denominator, resulting in 1.2 - 0.6i.
Step-by-step explanation:
To write the quotient 9/6+3i in the form a+bi, we need to eliminate the imaginary part from the denominator.
To do this, we multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 6+3i is 6-3i.
So we multiply (9/6+3i) × (6-3i)/(6-3i).
First, we find the product of the numerators: 9 × (6-3i) = 54 - 27i.
Then, we use the fact that when the product of two complex numbers in the form (a + ib)(a - ib) is calculated, the result is a real number as per the rule A* A = a² + b².
Thus, we multiply 6+3i and 6-3i: (6+3i)(6-3i) = 6² - (3i)²
= 36 + 9
= 45.
Finally, we divide the product of the numerators by the product of the denominators:
(54 - 27i)/45 = 54/45 - 27i/45.
To simplify, we divide both terms by 45: 1.2 - 0.6i.
So the quotient in the form a+bi is 1.2 - 0.6i.