Multiply $(9-4 i)(2+5 i)$

The product of the complex numbers ( 9 − 4 i ) ( 2 + 5 i ) is calculated by using the distributive property, leading to a final result of 38 + 37 i . This involves expanding the expression, combining like terms, and substituting i 2 = − 1 . The final answer is 38 + 37 i .
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Answered by Anonymous | 2025-07-04

Multiply the complex numbers using the distributive property: ( 9 − 4 i ) ( 2 + 5 i ) = 18 + 45 i − 8 i − 20 i 2 .
Simplify the expression by combining like terms: 18 + 37 i − 20 i 2 .
Substitute i 2 = − 1 : 18 + 37 i + 20 .
Combine the real terms to get the final answer: 38 + 37 i ​ .

Explanation

Understanding the Problem We are asked to multiply two complex numbers, ( 9 − 4 i ) and ( 2 + 5 i ) . Recall that i is the imaginary unit, defined as i 2 = − 1 . Our goal is to express the product in the form a + bi , where a and b are real numbers.

Expanding the Product To multiply the two complex numbers, we use the distributive property (also known as the FOIL method): ( 9 − 4 i ) ( 2 + 5 i ) = 9 ( 2 ) + 9 ( 5 i ) − 4 i ( 2 ) − 4 i ( 5 i ) = 18 + 45 i − 8 i − 20 i 2

Simplifying the Expression Now, we simplify the expression by combining like terms and substituting i 2 = − 1 :
18 + 45 i − 8 i − 20 i 2 = 18 + ( 45 − 8 ) i − 20 ( − 1 ) = 18 + 37 i + 20

Final Result Finally, we combine the real terms to obtain the result in the form a + bi :
18 + 37 i + 20 = ( 18 + 20 ) + 37 i = 38 + 37 i


Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The voltage and current in a circuit can be represented as complex numbers, and the impedance of circuit elements (resistors, capacitors, and inductors) can also be represented as complex numbers. By using complex numbers, engineers can easily calculate the behavior of AC circuits.

Answered by GinnyAnswer | 2025-07-04