For the imaginary unit i, where i^2 = -1, which of the following complex numbers is equal … — ACT

For the imaginary unit $i$ , where $i^2 = -1$ , which of the following complex numbers is equal to $\frac{8 + i}{3 + 2i}$ ?

The complex number

2 - i

To divide two complex numbers, we multiply both the numerator and denominator by the complex conjugate of the denominator, which is

3 - 2i

. Expanding the numerator gives

(8 + i)(3 - 2i) = 24 - 16i + 3i - 2i^2 = 26 - 13i

because

i^2 = -1

. Expanding the denominator yields

(3 + 2i)(3 - 2i) = 9 - 4i^2 = 9 + 4 = 13

. Dividing the terms of the numerator by the denominator gives

\frac{26 - 13i}{13} = 2 - i

Multiply the numerator and the denominator of the fraction by the complex conjugate of the denominator.

\frac{8 + i}{3 + 2i} \cdot \frac{3 - 2i}{3 - 2i} = \frac{(8 + i)(3 - 2i)}{(3 + 2i)(3 - 2i)}

Multiplying by the conjugate rationalizes the denominator, converting it into a real number.

Expand the numerator and the denominator using binomial multiplication.

Numerator:

(8 + i)(3 - 2i) = 24 - 16i + 3i - 2i^2

Denominator:

(3 + 2i)(3 - 2i) = 9 - 6i + 6i - 4i^2 = 9 - 4i^2

Distribute each term in the first binomial to each term in the second binomial.

Substitute

i^2 = -1

and simplify both expressions.

Numerator:

24 - 13i - 2(-1) = 24 - 13i + 2 = 26 - 13i

Denominator:

9 - 4(-1) = 9 + 4 = 13

The definition of the imaginary unit is

i^2 = -1

Divide each term of the simplified numerator by the simplified denominator.

\frac{26 - 13i}{13} = \frac{26}{13} - \frac{13i}{13} = 2 - i

Separate the real and imaginary parts to write the complex number in standard form

a + bi

Division of complex numbers using the complex conjugate of the denominator.

Instead of dividing directly, let the result be

x + yi

. Then

(x + yi)(3 + 2i) = 8 + i

. Expanding this gives

(3x - 2y) + (2x + 3y)i = 8 + i

. Equating the real and imaginary parts gives the system of equations

3x - 2y = 8

and

2x + 3y = 1

. Solving this system yields

x = 2

and

y = -1

, which corresponds to the complex number

2 - i

Estimated Time:1m 30s

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