Evaluate the expression $(-3-2 i)(-4+2 i)$ and write the result in the form $a + b i$.

Question

ms1313 ms1313

02-11-2026
Mathematics

Answered

Evaluate the expression $(-3-2 i)(-4+2 i)$ and write the result in the form $a + b i$.

Verified Answer :

Related Questions

Choose the correct verb to complete the sentence Los estudiantes la tarea

Why is it important to use constants in an experiment

Where does a peptide bond form

give a number you can cube to give an answer bigger than 600 but smaller than 800 Please give the correct answer and it needs to be easy to understand thx D

a minivan can hold up to 7 people How many minivans are needed to take 45 people to a basketball game

multiplies to 294 but adds to 7

During an experiment which factors are observed and measured

How do you turn a remainder into a decimal

Mrs Lewis will put a fence around her rectangular garden The length of the garden is 9 58 yards The width of the garden is 5 14 yards How many yards of fencing

there are 600 pupils in a school the ratio of boys to girls in this school is 35How many girls and how many boys are in the school

GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T15:14:45-05:00

Understand that multiplying two complex numbers involves the distributive property (FOIL) and the definition $i^2 = -1$.
Multiply the terms to get $12 - 6i + 8i - 4i^2$.
Simplify the imaginary components ($2i$) and convert the imaginary unit square ($-4i^2 = 4$).
Add the real parts ($12+4=16$) to arrive at the result $\boxed{16+2i}$.

Explanation

Analysis of Complex Multiplication

We are tasked with evaluating the product of two complex numbers: $(-3 - 2i)$ and $(-4 + 2i)$. A complex number is written in the standard form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, defined by the property $i^2 = -1$. Our goal is to perform the multiplication and simplify the expression into the same standard form. To multiply two binomials of this type, we can use the distributive property, commonly known as the FOIL method (First, Outer, Inner, Last).

Applying the Distributive Property

To evaluate the product $(-3 - 2i)(-4 + 2i)$, we apply the FOIL method as follows:

First: $(-3) \times (-4) = 12$
Outer: $(-3) \times (2i) = -6i$
Inner: $(-2i) \times (-4) = 8i$
Last: $(-2i) \times (2i) = -4i^2$

Summing these terms together, we have:
\[12 - 6i + 8i - 4i^2\]

Simplifying and Final Result

Now, we combine the terms and simplify. First, combine the imaginary parts $-6i$ and $8i$, which gives $2i$.
Next, we substitute the property $i^2 = -1$ into the final term:
\[-4i^2 = -4(-1) = 4\]
Now, the expression becomes:
\[12 + 2i + 4\]
Finally, we add the real parts $12$ and $4$ to obtain the standard form:
\[16 + 2i\]
Thus, the product of the two complex numbers is $16 + 2i$.

Examples

Complex number multiplication is used in various fields like electrical engineering to represent AC signals. For example, if an electrical current $I = -3 - 2i$ flows through a component with impedance $Z = -4 + 2i$, the voltage $V$ across the component can be calculated using the formula $V = I \times Z$. By multiplying these complex numbers, engineers determine the voltage's magnitude and phase shift, which is essential for analyzing circuit performance and stability.