Expanding a Squared Binomial Expression

Oct 13, 2025 by magicalceylontours.com 85 views

Introduction

In mathematics, expanding and combining like terms is a fundamental concept that helps us simplify complex algebraic expressions. It is an essential skill that is used extensively in various branches of mathematics, including algebra, geometry, and calculus. In this article, we will delve into the world of expanding and combining like terms, focusing on the concept of expanding a squared binomial expression.

What are Like Terms?

Before we dive into the world of expanding and combining like terms, let's first understand what like terms are. Like terms are terms that have the same variable(s) raised to the same power. For example, in the expression $2x^2 + 3x^2$ , the terms $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Expanding a Squared Binomial Expression

A squared binomial expression is an expression of the form $(a + b)^2$ . To expand this expression, we can use the formula $(a + b)^2 = a^2 + 2ab + b^2$ . Let's apply this formula to the expression $\left(4+7a^2\right)^2$ .

Step 1: Apply the Formula

To expand the expression $\left(4+7a^2\right)^2$ , we can apply the formula $(a + b)^2 = a^2 + 2ab + b^2$ . In this case, $a = 4$ and $b = 7a^2$ . Substituting these values into the formula, we get:

$\left(4+7a^2\right)^2 = (4)^2 + 2(4)(7a^2) + (7a^2)^2$

Step 2: Simplify the Expression

Now, let's simplify the expression by evaluating the terms:

$\left(4+7a^2\right)^2 = 16 + 56a^2 + 49a^4$

Combining Like Terms

In the expression $16 + 56a^2 + 49a^4$ , we have three terms: $16$ , $56a^2$ , and $49a^4$ . These terms are like terms because they all have the same variable(s) raised to the same power. To combine like terms, we can add or subtract the coefficients of the like terms.

Step 1: Identify the Like Terms

In the expression $16 + 56a^2 + 49a^4$ , the like terms are $16$ , $56a^2$ , and $49a^4$ . These terms have the same variable(s) raised to the same power.

Step 2: Combine the Like Terms

To combine the like terms, we can add or subtract the coefficients of the like terms. In this case, we can add the coefficients of the like terms:

$16 + 56a^2 + 49a^4 = 16 + 56a^2 + 49a^4$

Conclusion

In this article, we have explored the concept of expanding and combining like terms, focusing on the concept of expanding a squared binomial expression. We have applied the formula $(a + b)^2 = a^2 + 2ab + b^2$ to expand the expression $\left(4+7a^2\right)^2$ and simplified the expression to obtain the result $16 + 56a^2 + 49a^4$ . We have also combined like terms in the expression $16 + 56a^2 + 49a^4$ to obtain the final result.

Examples and Exercises

Here are some examples and exercises to help you practice expanding and combining like terms:

Example 1

Expand the expression $(x + 2)^2$ .

Solution

To expand the expression $(x + 2)^2$ , we can apply the formula $(a + b)^2 = a^2 + 2ab + b^2$ . In this case, $a = x$ and $b = 2$ . Substituting these values into the formula, we get:

$(x + 2)^2 = (x)^2 + 2(x)(2) + (2)^2$

Simplifying the expression, we get:

$(x + 2)^2 = x^2 + 4x + 4$

Example 2

Combine the like terms in the expression $3x^2 + 2x^2 + 5x^2$ .

Solution

To combine the like terms in the expression $3x^2 + 2x^2 + 5x^2$ , we can add the coefficients of the like terms:

$3x^2 + 2x^2 + 5x^2 = 10x^2$

Tips and Tricks

Here are some tips and tricks to help you expand and combine like terms:

Always apply the formula $(a + b)^2 = a^2 + 2ab + b^2$ when expanding a squared binomial expression.
Identify the like terms in an expression and combine them by adding or subtracting the coefficients.
Simplify the expression by evaluating the terms and combining like terms.

Conclusion

Introduction

In our previous article, we explored the concept of expanding and combining like terms, focusing on the concept of expanding a squared binomial expression. In this article, we will answer some frequently asked questions about expanding and combining like terms.

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

A: The formula for expanding a squared binomial expression is $(a + b)^2 = a^2 + 2ab + b^2$ .

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

A: To apply the formula, you need to identify the values of $a$ and $b$ in the expression. Then, substitute these values into the formula and simplify the expression.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, in the expression $2x^2 + 3x^2$ , the terms $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, in the expression $3x^2 + 2x^2 + 5x^2$ , you can combine the like terms by adding the coefficients:

$3x^2 + 2x^2 + 5x^2 = 10x^2$

Q: What is the difference between expanding and combining like terms?

A: Expanding a squared binomial expression involves applying the formula $(a + b)^2 = a^2 + 2ab + b^2$ to simplify the expression. Combining like terms involves adding or subtracting the coefficients of the like terms to simplify the expression.

Q: Can I use the formula to expand expressions that are not squared binomials?

A: No, the formula $(a + b)^2 = a^2 + 2ab + b^2$ is only applicable to squared binomial expressions. If you have an expression that is not a squared binomial, you will need to use a different method to simplify it.

Q: How do I simplify expressions with multiple variables?

A: To simplify expressions with multiple variables, you need to identify the like terms and combine them by adding or subtracting the coefficients. For example, in the expression $2x^2y + 3x^2y + 5x^2y$ , you can combine the like terms by adding the coefficients:

$2x^2y + 3x^2y + 5x^2y = 10x^2y$

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions. However, it's always a good idea to check your work by hand to make sure you understand the process.

Q: What are some common mistakes to avoid when expanding and combining like terms?

A: Some common mistakes to avoid when expanding and combining like terms include:

Forgetting to apply the formula when expanding a squared binomial expression
Not identifying the like terms in an expression
Not combining the like terms correctly
Not simplifying the expression by evaluating the terms and combining like terms

Conclusion

In conclusion, expanding and combining like terms is an essential skill in mathematics that helps us simplify complex algebraic expressions. By applying the formula $(a + b)^2 = a^2 + 2ab + b^2$ and combining like terms, we can simplify expressions and solve problems in various branches of mathematics. With practice and patience, you can master the art of expanding and combining like terms and become proficient in solving complex algebraic expressions.

Additional Resources

Here are some additional resources to help you learn more about expanding and combining like terms:

Khan Academy: Expanding and Combining Like Terms
Mathway: Expanding and Combining Like Terms
Wolfram Alpha: Expanding and Combining Like Terms

Practice Problems

Here are some practice problems to help you practice expanding and combining like terms:

Problem 1

Expand the expression $(x + 2)^2$ .

Solution

To expand the expression $(x + 2)^2$ , we can apply the formula $(a + b)^2 = a^2 + 2ab + b^2$ . In this case, $a = x$ and $b = 2$ . Substituting these values into the formula, we get:

$(x + 2)^2 = (x)^2 + 2(x)(2) + (2)^2$

Simplifying the expression, we get:

$(x + 2)^2 = x^2 + 4x + 4$

Problem 2

Combine the like terms in the expression $3x^2 + 2x^2 + 5x^2$ .

Solution

To combine the like terms in the expression $3x^2 + 2x^2 + 5x^2$ , we can add the coefficients of the like terms:

$3x^2 + 2x^2 + 5x^2 = 10x^2$

Answer Key

Here is the answer key for the practice problems:

Problem 1

$(x + 2)^2 = x^2 + 4x + 4$

Problem 2

$3x^2 + 2x^2 + 5x^2 = 10x^2$