Multiplying and Dividing Complex Numbers Worksheet - Lesson 2 with practice problems and solutions.

Problem Analysis:

The task involves simplifying algebraic expressions and solving equations. Let's break it down step by step.

#### Given Information:
1. Expressions to Simplify:
- Expression 1: \( (3x + 5) - (4x - 2) \)
- Expression 2: \( x^2 - 8x + 40 - (x^2 - 10) \)
- Expression 3: \( A = (2x - 3)^2 \)

2. Equation to Solve:
- Equation: \( B = 134 - C \)

3. Instructions:
- Simplify the given expressions.
- Solve for \( B \) in terms of \( C \).

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Step-by-Step Solution:

#### Expression 1: Simplify \( (3x + 5) - (4x - 2) \)

1. Distribute the negative sign through the second parenthesis:
\[
(3x + 5) - (4x - 2) = 3x + 5 - 4x + 2
\]

2. Combine like terms:
- Combine the \( x \)-terms: \( 3x - 4x = -x \)
- Combine the constant terms: \( 5 + 2 = 7 \)

Therefore:
\[
(3x + 5) - (4x - 2) = -x + 7
\]

Simplified Expression 1:
\[
\boxed{-x + 7}
\]

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#### Expression 2: Simplify \( x^2 - 8x + 40 - (x^2 - 10) \)

1. Distribute the negative sign through the second parenthesis:
\[
x^2 - 8x + 40 - (x^2 - 10) = x^2 - 8x + 40 - x^2 + 10
\]

2. Combine like terms:
- Combine the \( x^2 \)-terms: \( x^2 - x^2 = 0 \)
- Combine the \( x \)-terms: \( -8x \) (no other \( x \)-terms to combine)
- Combine the constant terms: \( 40 + 10 = 50 \)

Therefore:
\[
x^2 - 8x + 40 - (x^2 - 10) = -8x + 50
\]

Simplified Expression 2:
\[
\boxed{-8x + 50}
\]

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#### Expression 3: Simplify \( A = (2x - 3)^2 \)

1. Expand the square using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[
(2x - 3)^2 = (2x)^2 - 2(2x)(3) + 3^2
\]

2. Calculate each term:
- \( (2x)^2 = 4x^2 \)
- \( -2(2x)(3) = -12x \)
- \( 3^2 = 9 \)

Therefore:
\[
(2x - 3)^2 = 4x^2 - 12x + 9
\]

Simplified Expression 3:
\[
\boxed{4x^2 - 12x + 9}
\]

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#### Equation: Solve \( B = 134 - C \)

This equation is already in its simplest form, expressing \( B \) in terms of \( C \). No further simplification is needed.

Solution for the Equation:
\[
\boxed{B = 134 - C}
\]

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Final Answers:

1. Simplified Expression 1: \(\boxed{-x + 7}\)
2. Simplified Expression 2: \(\boxed{-8x + 50}\)
3. Simplified Expression 3: \(\boxed{4x^2 - 12x + 9}\)
4. Solution for the Equation: \(\boxed{B = 134 - C}\)