Factor $5x^2 - 2x + 5x - 2$ By Grouping.A) $(x+1)(5x-2)$ B) \$(x-1)(5x+2)$[/tex\] C) $(x-2)(5x+1)$ D) $(x+2)(5x-1)$
Introduction
In algebra, factorizing a quadratic expression is an essential skill that helps us simplify complex equations and solve problems more efficiently. One of the methods used to factorize a quadratic expression is by grouping. This method involves rearranging the terms of the quadratic expression to group them in a way that allows us to factor out common factors. In this article, we will learn how to factorize the quadratic expression $5x^2 - 2x + 5x - 2$ by grouping.
Understanding the Quadratic Expression
Before we proceed with factorizing the quadratic expression, let's first understand what it represents. The given quadratic expression is $5x^2 - 2x + 5x - 2$. This expression consists of four terms: $5x^2$, $-2x$, $5x$, and $-2$. Our goal is to factorize this expression by grouping the terms in a way that allows us to simplify it.
Step 1: Rearrange the Terms
To factorize the quadratic expression by grouping, we need to rearrange the terms in a way that allows us to group them together. Let's rearrange the terms as follows:
Step 2: Group the Terms
Now that we have rearranged the terms, let's group them together. We can group the first two terms ($5x^2$ and $5x$) and the last two terms ($-2x$ and $-2$) as follows:
Step 3: Factor Out Common Factors
Now that we have grouped the terms, let's factor out common factors from each group. From the first group ($5x^2 + 5x$), we can factor out $5x$ as follows:
From the second group ($-2x - 2$), we can factor out $-2$ as follows:
Step 4: Write the Factored Form
Now that we have factored out common factors from each group, let's write the factored form of the quadratic expression. We can combine the two groups as follows:
Step 5: Factor Out the Common Binomial
Now that we have written the factored form of the quadratic expression, let's factor out the common binomial ($x + 1$) from both groups. We can factor out $x + 1$ as follows:
Conclusion
In this article, we learned how to factorize the quadratic expression $5x^2 - 2x + 5x - 2$ by grouping. We rearranged the terms, grouped them together, factored out common factors, and wrote the factored form of the quadratic expression. The final answer is:
This is the correct answer among the options provided. The other options are:
These options are incorrect because they do not match the factored form of the quadratic expression that we obtained by grouping.
Final Answer
The final answer is:
(x + 1)(5x - 2)$<br/> **Factorizing a Quadratic Expression by Grouping: Q&A** ===================================================== **Introduction** --------------- In our previous article, we learned how to factorize a quadratic expression by grouping. We applied this method to the quadratic expression $5x^2 - 2x + 5x - 2$ and obtained the factored form $(x + 1)(5x - 2)$. In this article, we will answer some frequently asked questions about factorizing a quadratic expression by grouping. **Q: What is the first step in factorizing a quadratic expression by grouping?** --------------------------------------------------------- A: The first step in factorizing a quadratic expression by grouping is to rearrange the terms of the quadratic expression. This involves rearranging the terms in a way that allows us to group them together. **Q: How do I know which terms to group together?** ---------------------------------------------- A: To determine which terms to group together, we need to look for common factors among the terms. We can group the terms that have common factors together. **Q: What is the next step after grouping the terms?** ---------------------------------------------- A: After grouping the terms, the next step is to factor out common factors from each group. This involves factoring out the greatest common factor (GCF) from each group. **Q: Can I factor out common factors from both groups at the same time?** ---------------------------------------------------------------- A: No, you cannot factor out common factors from both groups at the same time. You need to factor out common factors from each group separately. **Q: What is the final step in factorizing a quadratic expression by grouping?** ---------------------------------------------------------------- A: The final step in factorizing a quadratic expression by grouping is to write the factored form of the quadratic expression. This involves combining the two groups and writing the factored form. **Q: Why is it important to factorize a quadratic expression by grouping?** ---------------------------------------------------------------- A: Factorizing a quadratic expression by grouping is important because it helps us simplify complex equations and solve problems more efficiently. It also helps us to identify the roots of the quadratic equation. **Q: Can I use the method of grouping to factorize any quadratic expression?** ---------------------------------------------------------------- A: Yes, you can use the method of grouping to factorize any quadratic expression. However, you need to make sure that the quadratic expression can be factored by grouping. **Q: What are some common mistakes to avoid when factorizing a quadratic expression by grouping?** --------------------------------------------------------------------------------------------- A: Some common mistakes to avoid when factorizing a quadratic expression by grouping include: * Not rearranging the terms correctly * Not factoring out common factors correctly * Not writing the factored form correctly **Q: How can I practice factorizing a quadratic expression by grouping?** ---------------------------------------------------------------- A: You can practice factorizing a quadratic expression by grouping by working on examples and exercises. You can also use online resources and practice tests to help you improve your skills. **Conclusion** ---------- In this article, we answered some frequently asked questions about factorizing a quadratic expression by grouping. We covered topics such as the first step in factorizing a quadratic expression by grouping, how to determine which terms to group together, and the final step in factorizing a quadratic expression by grouping. We also discussed some common mistakes to avoid and how to practice factorizing a quadratic expression by grouping. **Final Tips** ------------ * Make sure to rearrange the terms correctly before grouping them together. * Factor out common factors correctly from each group. * Write the factored form correctly. * Practice factorizing a quadratic expression by grouping regularly to improve your skills. **Final Answer** -------------- The final answer is: $(x + 1)(5x - 2)