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

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Introduction

Lucia, a diligent student, was working on a math problem involving the addition of two quadratic expressions. However, she arrived at an unexpected result, which led her to question her work. In this article, we will delve into Lucia's algebraic error and identify the mistake she made.

The Given Problem

Lucia was given the following problem:

(3x2+3x+5)+(7x2−9x+8)=10x2−12x+13\left(3 x^2+3 x+5\right)+\left(7 x^2-9 x+8\right)=10 x^2-12 x+13

Step-by-Step Analysis

To identify Lucia's error, let's analyze the problem step by step.

Step 1: Combine Like Terms

The first step in adding the two quadratic expressions is to combine like terms. Like terms are terms that have the same variable raised to the same power.

In this case, the like terms are:

  • 3x23 x^2 and 7x27 x^2
  • 3x3 x and −9x-9 x
  • 55 and 88

Step 2: Add the Coefficients

Now, let's add the coefficients of the like terms.

  • For the x2x^2 terms, we have 3x2+7x2=10x23 x^2 + 7 x^2 = 10 x^2
  • For the xx terms, we have 3x−9x=−6x3 x - 9 x = -6 x
  • For the constant terms, we have 5+8=135 + 8 = 13

Step 3: Write the Result

Now that we have combined the like terms and added the coefficients, we can write the result.

(3x2+3x+5)+(7x2−9x+8)=10x2−6x+13\left(3 x^2+3 x+5\right)+\left(7 x^2-9 x+8\right)=10 x^2-6 x+13

Conclusion

Lucia's error was that she combined the terms 3x23 x^2 and 7x27 x^2 incorrectly. She should have added the coefficients of the like terms, which would have resulted in 10x2−6x+1310 x^2 - 6 x + 13 instead of 10x2−12x+1310 x^2 - 12 x + 13.

Answer

The correct answer is:

B. She combined the terms 3x23 x^2 and 7x27 x^2 incorrectly.

Discussion

This problem highlights the importance of carefully combining like terms when adding or subtracting polynomials. It also demonstrates the need to double-check one's work to ensure that the result is correct.

Tips and Tricks

  • When combining like terms, make sure to add the coefficients of the like terms.
  • Double-check your work to ensure that the result is correct.
  • Use a calculator or a computer algebra system to check your work if you are unsure.

Real-World Applications

This problem has real-world applications in various fields, such as:

  • Engineering: When designing a system or a structure, engineers need to combine like terms to ensure that the system or structure is stable and functional.
  • Economics: When analyzing economic data, economists need to combine like terms to understand the relationships between different variables.
  • Science: When conducting scientific experiments, scientists need to combine like terms to understand the relationships between different variables.

Conclusion

In conclusion, Lucia's error was that she combined the terms 3x23 x^2 and 7x27 x^2 incorrectly. She should have added the coefficients of the like terms, which would have resulted in 10x2−6x+1310 x^2 - 6 x + 13 instead of 10x2−12x+1310 x^2 - 12 x + 13. This problem highlights the importance of carefully combining like terms when adding or subtracting polynomials.

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Introduction

In our previous article, we analyzed Lucia's algebraic error and identified the mistake she made. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the main error that Lucia made?

A: Lucia's main error was that she combined the terms 3x23 x^2 and 7x27 x^2 incorrectly. She should have added the coefficients of the like terms, which would have resulted in 10x2−6x+1310 x^2 - 6 x + 13 instead of 10x2−12x+1310 x^2 - 12 x + 13.

Q: What is the importance of combining like terms?

A: Combining like terms is an essential step in adding or subtracting polynomials. It helps to simplify the expression and make it easier to work with. In this case, Lucia's error was that she did not combine the like terms correctly, which led to an incorrect result.

Q: How can I avoid making the same mistake as Lucia?

A: To avoid making the same mistake as Lucia, make sure to carefully combine like terms when adding or subtracting polynomials. Double-check your work to ensure that the result is correct. Use a calculator or a computer algebra system to check your work if you are unsure.

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

A: Combining like terms has real-world applications in various fields, such as:

  • Engineering: When designing a system or a structure, engineers need to combine like terms to ensure that the system or structure is stable and functional.
  • Economics: When analyzing economic data, economists need to combine like terms to understand the relationships between different variables.
  • Science: When conducting scientific experiments, scientists need to combine like terms to understand the relationships between different variables.

Q: How can I practice combining like terms?

A: To practice combining like terms, try the following exercises:

  • Add or subtract polynomials with like terms.
  • Use a calculator or a computer algebra system to check your work.
  • Practice combining like terms with different variables and coefficients.

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

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

  • Failing to combine like terms correctly.
  • Adding or subtracting coefficients incorrectly.
  • Not double-checking your work.

Q: How can I use technology to help me with combining like terms?

A: You can use technology, such as calculators or computer algebra systems, to help you with combining like terms. These tools can help you to:

  • Check your work for errors.
  • Simplify complex expressions.
  • Explore different scenarios and variables.

Conclusion

In conclusion, combining like terms is an essential step in adding or subtracting polynomials. By carefully combining like terms and double-checking your work, you can avoid making mistakes like Lucia and ensure that your results are accurate. Remember to practice combining like terms with different variables and coefficients, and use technology to help you with the process.

Tips and Tricks

  • Always double-check your work when combining like terms.
  • Use a calculator or a computer algebra system to check your work.
  • Practice combining like terms with different variables and coefficients.

Real-World Applications

Combining like terms has real-world applications in various fields, such as:

  • Engineering: When designing a system or a structure, engineers need to combine like terms to ensure that the system or structure is stable and functional.
  • Economics: When analyzing economic data, economists need to combine like terms to understand the relationships between different variables.
  • Science: When conducting scientific experiments, scientists need to combine like terms to understand the relationships between different variables.

Conclusion

In conclusion, combining like terms is an essential step in adding or subtracting polynomials. By carefully combining like terms and double-checking your work, you can avoid making mistakes like Lucia and ensure that your results are accurate. Remember to practice combining like terms with different variables and coefficients, and use technology to help you with the process.