Problem 63 Multiply and simplify. Write eac... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 7: Problem 63

Multiply and simplify. Write each answer in the form $a+b i$. $$(-2+3 i)^{2}$$

Short Answer

Expert verified

-5 - 12i

Step by step solution

- Expand the Binomial

Use the binomial theorem to expand $(-2 + 3i)^2$. We can expand it as follows: $(-2 + 3i)^2 = (-2)^2 + 2*(-2)*(3i) + (3i)^2$.

- Calculate Each Term

Compute each term from the expansion: $ (-2)^2 = 4 $, $ 2*(-2)*(3i) = -12i $ and $ (3i)^2 = 9i^2 $.

- Simplify the Imaginary Term

Recall that $i^2 = -1$ and simplify the last term: $(3i)^2 = 9i^2 = 9*(-1) = -9$.

- Combine the Real and Imaginary Parts

Add the real parts and the imaginary parts separately: $ 4 - 9 = -5 $ and the only imaginary part is $-12i$. The result is $-5 - 12i$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expansion

When we talk about binomial expansion, we mean expressing a binomial raised to a power in expanded form. A binomial is a polynomial with two terms, like $(-2 + 3i)$. In this exercise, we need to expand $(-2 + 3i)^2$.

Use the binomial theorem which states $(a + b)^n$ can be expanded using the formula:
\[a^2 + 2ab + b^2\]

For this problem:
$a = -2$ and $b = 3i$, and we put them into the formula to get:
$(-2)^2 + 2(-2)(3i) + (3i)^2$.
Always remember to simplify each term one by one.

Imaginary Unit

The imaginary unit, denoted as $i$, is an important concept in complex numbers. It is defined as the square root of -1. This means $i^2 = -1$.

Using this property, any time you see $i^2$ in your expansion or calculations, you can replace it with -1.

In our exercise, when we get $(3i)^2 = 9i^2$, we know that this can be simplified to $9(-1) = -9$. The key is recognizing how $i^2$ transforms into -1 for easier simplification.

Complex Numbers

Complex numbers have two parts: a real part and an imaginary part. They are generally expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

In our problem, we've expanded and simplified$(-2+3i)^2$. After simplifying, we combine the real parts together and the imaginary parts together. We end up with $-5 - 12i$.

This is now in the correct form $a + bi$, where $a = -5$ and $b = -12$, giving the final answer as $-5 -12i$. Always be sure your final answer splits into the real and imaginary components for correct representation.

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91影视

Multiply and simplify. Write each answer in the form \(a+b i\). $$(-2+3 i)^{2}$$

Short Answer

Step by step solution

- Expand the Binomial

- Calculate Each Term

- Simplify the Imaginary Term

- Combine the Real and Imaginary Parts

Key Concepts

Binomial Expansion

Imaginary Unit

Complex Numbers

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