Expand The Expression: { (2x - 3y)^2$}$

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Introduction

In algebra, expanding an expression involves expressing it as a sum of simpler terms. In this article, we will focus on expanding the expression {(2x - 3y)^2$}$. This is a quadratic expression in two variables, x and y, and it involves the square of a binomial. We will use the formula for expanding a squared binomial to simplify the expression.

The Formula for Expanding a Squared Binomial

The formula for expanding a squared binomial is:

${(a - b)^2 = a^2 - 2ab + b^2}$

This formula can be used to expand any squared binomial of the form ${(a - b)^2}$. In our case, we have ${(2x - 3y)^2}$, so we can use the formula to expand it.

Expanding the Expression

Using the formula for expanding a squared binomial, we can expand the expression ${(2x - 3y)^2}$ as follows:

${(2x - 3y)^2 = (2x)^2 - 2(2x)(3y) + (3y)^2}$

Expanding each term, we get:

${(2x)^2 = 4x^2}$

${-2(2x)(3y) = -12xy}$

${(3y)^2 = 9y^2}$

So, the expanded expression is:

${(2x - 3y)^2 = 4x^2 - 12xy + 9y^2}$

Simplifying the Expression

The expanded expression ${4x^2 - 12xy + 9y^2}$ can be simplified further by combining like terms. In this case, there are no like terms, so the expression is already simplified.

Conclusion

In this article, we expanded the expression {(2x - 3y)^2$}$ using the formula for expanding a squared binomial. We simplified the expression by combining like terms and obtained the final result: ${4x^2 - 12xy + 9y^2}$. This expression can be used in various mathematical applications, such as solving systems of equations or finding the area of a region.

Example Use Cases

The expression ${4x^2 - 12xy + 9y^2}$ can be used in various mathematical applications, such as:

• Solving Systems of Equations: The expression can be used to solve systems of equations involving quadratic terms.
• Finding the Area of a Region: The expression can be used to find the area of a region bounded by a quadratic curve.
• Optimization Problems: The expression can be used to solve optimization problems involving quadratic functions.

Tips and Tricks

When expanding a squared binomial, it's essential to use the formula carefully and to simplify the expression by combining like terms. Additionally, it's helpful to use algebraic manipulations, such as factoring and canceling, to simplify the expression further.

Common Mistakes

When expanding a squared binomial, it's common to make mistakes, such as:

• Forgetting to use the formula: Failing to use the formula for expanding a squared binomial can lead to incorrect results.
• Not simplifying the expression: Failing to simplify the expression by combining like terms can lead to unnecessary complexity.
• Making algebraic errors: Making algebraic errors, such as incorrect multiplication or addition, can lead to incorrect results.

Conclusion

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Introduction

In our previous article, we expanded the expression {(2x - 3y)^2$}$ using the formula for expanding a squared binomial. In this article, we will answer some frequently asked questions about expanding expressions like this one.

Q: What is the formula for expanding a squared binomial?

A: The formula for expanding a squared binomial is:

${(a - b)^2 = a^2 - 2ab + b^2}$

This formula can be used to expand any squared binomial of the form ${(a - b)^2}$.

Q: How do I use the formula to expand a squared binomial?

A: To use the formula to expand a squared binomial, simply substitute the values of a and b into the formula. For example, if we want to expand ${(2x - 3y)^2}$, we would substitute a = 2x and b = 3y into the formula.

Q: What is the expanded form of {(2x - 3y)^2$}$?

A: The expanded form of {(2x - 3y)^2$}$ is:

${(2x - 3y)^2 = 4x^2 - 12xy + 9y^2}$

Q: How do I simplify the expanded expression?

A: To simplify the expanded expression, combine like terms. In this case, there are no like terms, so the expression is already simplified.

Q: What are some common mistakes to avoid when expanding a squared binomial?

A: Some common mistakes to avoid when expanding a squared binomial include:

• Forgetting to use the formula: Failing to use the formula for expanding a squared binomial can lead to incorrect results.
• Not simplifying the expression: Failing to simplify the expression by combining like terms can lead to unnecessary complexity.
• Making algebraic errors: Making algebraic errors, such as incorrect multiplication or addition, can lead to incorrect results.

Q: How do I apply the formula for expanding a squared binomial to real-world problems?

A: The formula for expanding a squared binomial can be applied to a variety of real-world problems, such as:

• Solving Systems of Equations: The formula can be used to solve systems of equations involving quadratic terms.
• Finding the Area of a Region: The formula can be used to find the area of a region bounded by a quadratic curve.
• Optimization Problems: The formula can be used to solve optimization problems involving quadratic functions.

Q: What are some tips and tricks for expanding a squared binomial?

A: Some tips and tricks for expanding a squared binomial include:

• Use the formula carefully: Make sure to use the formula correctly and to simplify the expression by combining like terms.
• Simplify the expression: Simplify the expression by combining like terms to avoid unnecessary complexity.
• Check your work: Check your work carefully to avoid making algebraic errors.

Conclusion

In conclusion, expanding the expression {(2x - 3y)^2$}$ using the formula for expanding a squared binomial is a straightforward process. By following the formula and simplifying the expression by combining like terms, we can obtain the final result: ${4x^2 - 12xy + 9y^2}$. This expression can be used in various mathematical applications, such as solving systems of equations or finding the area of a region.