Simplify The Expression:$\[(x - (2 - 3i))(x - (2 + 3i))\\]

$$

Introduction

In this article, we will simplify the given expression $(x - (2 - 3i))(x - (2 + 3i))$. This involves using the distributive property and the difference of squares formula to simplify the expression. We will also explore the concept of complex conjugates and their role in simplifying expressions.

Understanding Complex Conjugates

Before we dive into simplifying the expression, let's understand what complex conjugates are. Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. In this case, the complex conjugates are $2 - 3i$ and $2 + 3i$. These two numbers are conjugates of each other because they have the same real part (2) but opposite imaginary parts (-3i and +3i).

Simplifying the Expression

To simplify the expression, we will use the distributive property and the difference of squares formula. The distributive property states that for any real numbers a, b, and c, we have:

a(b + c) = ab + ac

The difference of squares formula states that for any real numbers a and b, we have:

a^2 - b^2 = (a - b)(a + b)

Using the distributive property, we can expand the expression as follows:

(x - (2 - 3i))(x - (2 + 3i)) = x(x - (2 + 3i)) - (2 - 3i)(x - (2 + 3i))

Applying the Distributive Property

Now, let's apply the distributive property to each term:

x(x - (2 + 3i)) = x^2 - 2x - 3xi

-(2 - 3i)(x - (2 + 3i)) = -2x + 4 + 6i + 3xi - 3xi - 9i^2

Simplifying the Expression Further

Now, let's simplify the expression further by combining like terms:

x^2 - 2x - 3xi - 2x + 4 + 6i + 3xi - 3xi - 9i^2

Using the Difference of Squares Formula

We can simplify the expression further by using the difference of squares formula. Recall that i^2 = -1. Therefore, we can rewrite the expression as:

x^2 - 4x + 4 + 6i - 9(-1)

Simplifying the Expression

Now, let's simplify the expression further by combining like terms:

x^2 - 4x + 4 + 6i + 9

Final Simplification

The final simplified expression is:

x^2 - 4x + 13 + 6i

Conclusion

In this article, we simplified the expression $(x - (2 - 3i))(x - (2 + 3i))$ using the distributive property and the difference of squares formula. We also explored the concept of complex conjugates and their role in simplifying expressions. The final simplified expression is $x^2 - 4x + 13 + 6i$.

Frequently Asked Questions

• What are complex conjugates? Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts.
• How do you simplify an expression using complex conjugates? To simplify an expression using complex conjugates, you can use the distributive property and the difference of squares formula.
• What is the difference of squares formula? The difference of squares formula states that for any real numbers a and b, we have: a^2 - b^2 = (a - b)(a + b)

Further Reading

• Complex Numbers: A Comprehensive Guide
• Simplifying Expressions with Complex Conjugates
• The Distributive Property: A Guide to Simplifying Expressions

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Simplifying Expressions with Complex Conjugates" by Khan Academy
• [3] "The Distributive Property" by Purplemath
  

$$

Introduction

In our previous article, we simplified the expression $(x - (2 - 3i))(x - (2 + 3i))$ using the distributive property and the difference of squares formula. We also explored the concept of complex conjugates and their role in simplifying expressions. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What are complex conjugates?

A: Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. In this case, the complex conjugates are $2 - 3i$ and $2 + 3i$.

Q: How do you simplify an expression using complex conjugates?

A: To simplify an expression using complex conjugates, you can use the distributive property and the difference of squares formula. The distributive property states that for any real numbers a, b, and c, we have:

a(b + c) = ab + ac

The difference of squares formula states that for any real numbers a and b, we have:

a^2 - b^2 = (a - b)(a + b)

Q: What is the difference of squares formula?

A: The difference of squares formula states that for any real numbers a and b, we have:

a^2 - b^2 = (a - b)(a + b)

Q: How do you apply the distributive property to simplify an expression?

A: To apply the distributive property, you can multiply each term in the expression by the other term. For example, in the expression $(x - (2 - 3i))(x - (2 + 3i))$, you can multiply the first term $x$ by the second term $x - (2 + 3i)$ and then multiply the second term $-(2 - 3i)$ by the second term $x - (2 + 3i)$.

Q: What is the final simplified expression?

A: The final simplified expression is $x^2 - 4x + 13 + 6i$.

Q: Can you provide an example of how to simplify an expression using complex conjugates?

A: Yes, let's consider the expression $(x - (3 + 4i))(x - (3 - 4i))$. To simplify this expression, you can use the distributive property and the difference of squares formula. First, multiply the first term $x$ by the second term $x - (3 - 4i)$:

x(x - (3 - 4i)) = x^2 - 3x + 4xi

Then, multiply the second term $-(3 + 4i)$ by the second term $x - (3 - 4i)$:

-(3 + 4i)(x - (3 - 4i)) = -3x + 9 + 12i + 4xi - 12i - 16i^2

Now, simplify the expression by combining like terms:

x^2 - 3x + 4xi - 3x + 9 + 12i + 4xi - 12i - 16i^2

Recall that i^2 = -1. Therefore, we can rewrite the expression as:

x^2 - 6x + 9 + 8i + 16

Finally, simplify the expression by combining like terms:

x^2 - 6x + 25 + 8i

Q: Can you provide a list of common complex conjugate pairs?

A: Yes, here are some common complex conjugate pairs:

• 2 - 3i and 2 + 3i
• 3 + 4i and 3 - 4i
• 1 - 2i and 1 + 2i
• 4 + 5i and 4 - 5i

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions using complex conjugates. We also provided an example of how to simplify an expression using complex conjugates and a list of common complex conjugate pairs.

Frequently Asked Questions

• What are complex conjugates?
• How do you simplify an expression using complex conjugates?
• What is the difference of squares formula?
• How do you apply the distributive property to simplify an expression?
• What is the final simplified expression?
• Can you provide an example of how to simplify an expression using complex conjugates?
• Can you provide a list of common complex conjugate pairs?

Further Reading

• Complex Numbers: A Comprehensive Guide
• Simplifying Expressions with Complex Conjugates
• The Distributive Property: A Guide to Simplifying Expressions

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Simplifying Expressions with Complex Conjugates" by Khan Academy
• [3] "The Distributive Property" by Purplemath