Lucia Finds That $\left(3x^2 + 3x + 5\right) + \left(7x^2 - 9x + 8\right) = 10x^2 - 12x + 13$. What Error Did Lucia Make?A. She Found The Difference Instead Of The Sum.B. She Combined The Terms $3x^2$ And $7x^2$

Introduction

In algebra, combining like terms is a fundamental concept that helps simplify expressions and solve equations. However, even with the best intentions, mistakes can occur. In this article, we will delve into the error made by Lucia when she combined two quadratic expressions.

The Given Expression

Lucia was given the expression $\left(3x^2 + 3x + 5\right) + \left(7x^2 - 9x + 8\right) = 10x^2 - 12x + 13$. Her task was to simplify the expression by combining like terms.

The Correct Approach

To combine like terms, we need to identify the terms with the same variable and exponent. In this case, the like terms are the quadratic terms ($x^2$), the linear terms ($x$), and the constant terms.

Step 1: Combine the Quadratic Terms

The quadratic terms are $3x^2$ and $7x^2$. To combine them, we add their coefficients:

$3x^2 + 7x^2 = 10x^2$

Step 2: Combine the Linear Terms

The linear terms are $3x$ and $-9x$. To combine them, we add their coefficients:

$3x - 9x = -6x$

Step 3: Combine the Constant Terms

The constant terms are $5$ and $8$. To combine them, we add their values:

$5 + 8 = 13$

The Correct Simplified Expression

Now that we have combined the like terms, we can write the simplified expression:

$10x^2 - 6x + 13$

Lucia's Error

Lucia's error was not in combining the terms $3x^2$ and $7x^2$, but rather in not combining the linear terms $3x$ and $-9x$ correctly. She should have added their coefficients to get $-6x$, not $-12x$.

Conclusion

In conclusion, Lucia's error was not in finding the difference instead of the sum, but rather in not combining the linear terms correctly. By following the correct approach to combine like terms, we can simplify expressions and solve equations accurately.

Discussion Points

• What are some common mistakes that students make when combining like terms?
• How can we help students identify and correct their errors when combining like terms?
• What are some real-world applications of combining like terms in algebra?

Additional Resources

• Khan Academy: Combining Like Terms
• Mathway: Combining Like Terms
• Algebra.com: Combining Like Terms

Final Thoughts

Introduction

In our previous article, we analyzed the error made by Lucia when she combined two quadratic expressions. In this article, we will provide a Q&A guide to help you understand the concept of combining like terms and how to avoid common mistakes.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $3x^2$ and $7x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, to combine the terms $3x^2$ and $7x^2$, you would add their coefficients:

$3x^2 + 7x^2 = 10x^2$

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

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

• Not combining all like terms
• Combining terms with different variables or exponents
• Not adding or subtracting coefficients correctly
• Not simplifying the expression after combining like terms

Q: How can I simplify an expression after combining like terms?

A: To simplify an expression after combining like terms, you need to:

• Combine all like terms
• Simplify the expression by combining any remaining like terms
• Write the simplified expression in the correct format

Q: What are some real-world applications of combining like terms in algebra?

A: Combining like terms is used in many real-world applications, including:

• Simplifying expressions in physics and engineering
• Solving equations in finance and economics
• Analyzing data in statistics and data science
• Creating models in computer science and programming

Q: How can I practice combining like terms?

A: You can practice combining like terms by:

• Working through examples and exercises in your textbook or online resources
• Creating your own examples and exercises to practice
• Using online tools and calculators to check your work
• Asking a teacher or tutor for help and feedback

Q: What are some common errors to look out for when combining like terms?

A: Some common errors to look out for when combining like terms include:

• Not combining all like terms
• Combining terms with different variables or exponents
• Not adding or subtracting coefficients correctly
• Not simplifying the expression after combining like terms

Q: How can I avoid making mistakes when combining like terms?

A: To avoid making mistakes when combining like terms, you should:

• Read the problem carefully and understand what is being asked
• Identify all like terms and combine them correctly
• Simplify the expression after combining like terms
• Check your work and ask for help if you are unsure

Conclusion

Combining like terms is a fundamental concept in algebra that requires attention to detail and careful analysis. By understanding the correct approach to combining like terms and avoiding common mistakes, you can simplify expressions and solve equations accurately.