Understanding the Correct Format

Oct 13, 2025 by windailys.com 54 views

Introduction

Factoring is a fundamental concept in algebra, and it's essential to understand the correct procedures to factor expressions. In this article, we will explore a common mistake made by students when asked to factor a quadratic expression. We will examine the student's answer and identify the error, providing a clear explanation and examples to support our conclusion.

The Student's Answer

The student's answer to the factoring problem $$x^2 - 144$$ is $$(x - 12)(x - 12)$$. At first glance, this answer may seem correct, but it's essential to examine it more closely.

The Error in the Student's Answer

The student's answer is incorrect because it does not follow the correct format for factoring a quadratic expression. When factoring a quadratic expression of the form $$ax^2 + bx + c$$, we need to find two binomials that multiply together to give the original expression. In this case, the student's answer is $$(x - 12)(x - 12)$$, which is a repeated binomial.

Understanding the Correct Format

To factor a quadratic expression correctly, we need to find two binomials that multiply together to give the original expression. In this case, the correct format for factoring $$x^2 - 144$$ is $$(x - 12)(x + 12)$$. This is because $$(x - 12)(x + 12) = x^2 - 12x + 12x - 144 = x^2 - 144$$.

Why the Student's Answer is Incorrect

The student's answer is incorrect because it does not follow the correct format for factoring a quadratic expression. The repeated binomial $$(x - 12)(x - 12)$$ does not multiply together to give the original expression $$x^2 - 144$$. Instead, it would give $$(x - 12)^2 = x^2 - 24x + 144$$, which is a different expression altogether.

Conclusion

In conclusion, the student's answer $$(x - 12)(x - 12)$$ is incorrect because it does not follow the correct format for factoring a quadratic expression. The correct format for factoring $$x^2 - 144$$ is $$(x - 12)(x + 12)$$. This highlights the importance of understanding the correct procedures for factoring quadratic expressions and the need to carefully examine the student's answer to identify any errors.

Common Mistakes in Factoring

When factoring quadratic expressions, students often make common mistakes. Some of these mistakes include:

Incorrect signs: Students may incorrectly use a minus sign instead of a plus sign, or vice versa.
Repeated binomials: Students may incorrectly use a repeated binomial, such as $$(x - 12)(x - 12)$$, instead of the correct format.
Incorrect coefficients: Students may incorrectly use the wrong coefficients, such as using $$x^2 + 144$$ instead of $$x^2 - 144$$.

Examples of Correct Factoring

To illustrate the correct factoring of quadratic expressions, let's consider a few examples:

Example 1: Factor $$x^2 - 16$$.
- The correct format for factoring $$x^2 - 16$$ is $$(x - 4)(x + 4)$$.
- This is because $$(x - 4)(x + 4) = x^2 - 4x + 4x - 16 = x^2 - 16$$.
Example 2: Factor $$x^2 + 9$$.
- The correct format for factoring $$x^2 + 9$$ is $$(x - 3i)(x + 3i)$$.
- This is because $$(x - 3i)(x + 3i) = x^2 - 3ix + 3ix - 9i^2 = x^2 + 9$$.
Example 3: Factor $$x^2 - 25$$.
- The correct format for factoring $$x^2 - 25$$ is $$(x - 5)(x + 5)$$.
- This is because $$(x - 5)(x + 5) = x^2 - 5x + 5x - 25 = x^2 - 25$$.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra, and it's essential to understand the correct procedures to factor expressions. The student's answer $$(x - 12)(x - 12)$$ is incorrect because it does not follow the correct format for factoring a quadratic expression. The correct format for factoring $$x^2 - 144$$ is $$(x - 12)(x + 12)$$. This highlights the importance of understanding the correct procedures for factoring quadratic expressions and the need to carefully examine the student's answer to identify any errors.

Final Thoughts

Introduction

Factoring quadratic expressions is a fundamental concept in algebra, and it's essential to understand the correct procedures to factor expressions. In this article, we will provide a Q&A section to help clarify any doubts and provide additional examples to support our conclusion.

Q: What is the correct format for factoring a quadratic expression?

A: The correct format for factoring a quadratic expression of the form $$ax^2 + bx + c$$ is to find two binomials that multiply together to give the original expression.

Q: How do I identify the correct binomials to factor a quadratic expression?

A: To identify the correct binomials, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What if I have a quadratic expression with a negative constant term?

A: If you have a quadratic expression with a negative constant term, you need to use a minus sign in one of the binomials.

Q: Can I use a repeated binomial to factor a quadratic expression?

A: No, you cannot use a repeated binomial to factor a quadratic expression. A repeated binomial would give a different expression altogether.

Q: What if I have a quadratic expression with a coefficient of 1 on the x^2 term?

A: If you have a quadratic expression with a coefficient of 1 on the x^2 term, you can simply factor the expression by finding the two binomials that multiply together to give the original expression.

Q: Can I use the distributive property to factor a quadratic expression?

A: No, you cannot use the distributive property to factor a quadratic expression. The distributive property is used to multiply expressions, not to factor them.

Q: How do I know if a quadratic expression can be factored?

A: You can check if a quadratic expression can be factored by looking for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What if I have a quadratic expression that cannot be factored?

A: If you have a quadratic expression that cannot be factored, you can use other methods such as completing the square or using the quadratic formula to solve the equation.

Examples of Factoring Quadratic Expressions

To illustrate the correct factoring of quadratic expressions, let's consider a few examples:

Example 1: Factor $$x^2 - 16$$.
- The correct format for factoring $$x^2 - 16$$ is $$(x - 4)(x + 4)$$.
- This is because $$(x - 4)(x + 4) = x^2 - 4x + 4x - 16 = x^2 - 16$$.
Example 2: Factor $$x^2 + 9$$.
- The correct format for factoring $$x^2 + 9$$ is $$(x - 3i)(x + 3i)$$.
- This is because $$(x - 3i)(x + 3i) = x^2 - 3ix + 3ix - 9i^2 = x^2 + 9$$.
Example 3: Factor $$x^2 - 25$$.
- The correct format for factoring $$x^2 - 25$$ is $$(x - 5)(x + 5)$$.
- This is because $$(x - 5)(x + 5) = x^2 - 5x + 5x - 25 = x^2 - 25$$.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra, and it's essential to understand the correct procedures to factor expressions. By following the correct format for factoring quadratic expressions, we can ensure that our answers are accurate and complete. We hope this Q&A section has helped clarify any doubts and provided additional examples to support our conclusion.

Final Thoughts

Factoring quadratic expressions is a critical skill in algebra, and it's essential to understand the correct procedures to factor expressions. By carefully examining the student's answer and identifying any errors, we can provide a clear explanation and examples to support our conclusion. By following the correct format for factoring quadratic expressions, we can ensure that our answers are accurate and complete.