Divide: $\frac{-2+5 i}{-4-6 i}$. Write your answer in $a+b i$ form
Answer (2)
To divide the complex number − 4 − 6 i − 2 + 5 i , multiply by the conjugate − 4 + 6 i to eliminate the imaginary part in the denominator. This results in − 26 11 − 13 8 i , expressed in standard form. The process involves simplifying both the numerator and denominator through standard multiplication rules for complex numbers.
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Multiply the numerator and denominator by the conjugate of the denominator: − 4 − 6 i − 2 + 5 i = ( − 4 − 6 i ) ( − 4 + 6 i ) ( − 2 + 5 i ) ( − 4 + 6 i ) .
Simplify the numerator: ( − 2 + 5 i ) ( − 4 + 6 i ) = − 22 − 32 i .
Simplify the denominator: ( − 4 − 6 i ) ( − 4 + 6 i ) = 52 .
Divide and express in a + bi form: 52 − 22 − 32 i = − 26 11 − 13 8 i . The final answer is − 26 11 − 13 8 i .
Explanation
Problem Analysis We are asked to divide the complex number − 2 + 5 i by − 4 − 6 i and express the result in the standard form a + bi , where a and b are real numbers.
Finding the Conjugate To divide complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number c + d i is c − d i . Therefore, the conjugate of − 4 − 6 i is − 4 + 6 i .
Multiplying by the Conjugate Now, we multiply both the numerator and the denominator by the conjugate: − 4 − 6 i − 2 + 5 i = ( − 4 − 6 i ) ( − 4 + 6 i ) ( − 2 + 5 i ) ( − 4 + 6 i )
Simplifying the Numerator Let's multiply out the numerator: ( − 2 + 5 i ) ( − 4 + 6 i ) = ( − 2 ) ( − 4 ) + ( − 2 ) ( 6 i ) + ( 5 i ) ( − 4 ) + ( 5 i ) ( 6 i ) = 8 − 12 i − 20 i + 30 i 2 Since i 2 = − 1 , we have: 8 − 12 i − 20 i − 30 = − 22 − 32 i
Simplifying the Denominator Now, let's multiply out the denominator: ( − 4 − 6 i ) ( − 4 + 6 i ) = ( − 4 ) ( − 4 ) + ( − 4 ) ( 6 i ) + ( − 6 i ) ( − 4 ) + ( − 6 i ) ( 6 i ) = 16 − 24 i + 24 i − 36 i 2 Since i 2 = − 1 , we have: 16 + 36 = 52
Dividing and Simplifying Now, we divide the simplified numerator by the simplified denominator: 52 − 22 − 32 i = 52 − 22 − 52 32 i = 26 − 11 − 13 8 i Thus, the result is − 26 11 − 13 8 i .
Final Answer Therefore, the division of the given complex numbers in a + bi form is − 26 11 − 13 8 i ≈ − 0.423 − 0.615 i
Examples
Complex number division is used in electrical engineering to analyze AC circuits. Impedance, which is the opposition to current flow in an AC circuit, is a complex quantity. When analyzing circuits with multiple components, engineers often need to divide complex impedances to determine current flow or voltage drops. For example, calculating the current through a branch in a parallel AC circuit involves dividing the voltage (a complex number) by the impedance (another complex number) of that branch. The result provides both the magnitude and phase of the AC current.
