Factoring 2x² + 7x - 49 A Comprehensive Guide
Factoring quadratic expressions is a fundamental skill in algebra, and mastering it opens doors to solving a wide range of mathematical problems. In this article, we will delve into the process of factoring the quadratic expression 2x² + 7x - 49. We'll break down the steps, explore different approaches, and ultimately arrive at the correct factorization. Understanding the techniques involved in factoring quadratic expressions is crucial for success in algebra and beyond.
Understanding Quadratic Expressions
Before we dive into the specifics of factoring 2x² + 7x - 49, let's take a moment to understand the general form of a quadratic expression. A quadratic expression is a polynomial expression of the form ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The term ax² is the quadratic term, bx is the linear term, and c is the constant term.
In our case, the quadratic expression 2x² + 7x - 49 fits this form perfectly. Here, a = 2, b = 7, and c = -49. Recognizing these coefficients is the first step towards factoring the expression. Factoring a quadratic expression essentially means rewriting it as a product of two linear expressions. This process is the reverse of expanding two binomials, and it requires a keen eye for patterns and relationships between the coefficients.
Methods for Factoring Quadratic Expressions
There are several methods for factoring quadratic expressions, each with its own strengths and weaknesses. Some of the most common methods include:
- Trial and Error: This method involves making educated guesses about the factors and checking if they multiply to give the original expression. While it can be effective for simple quadratics, it can become time-consuming and frustrating for more complex expressions.
- Factoring by Grouping: This method involves splitting the middle term (bx) into two terms and then grouping terms to factor out common factors. It is a versatile method that can be applied to a wide range of quadratics.
- The AC Method: This method involves finding two numbers that multiply to ac and add up to b. These numbers are then used to split the middle term and factor by grouping. The AC method is particularly useful for quadratics where the coefficient of the quadratic term (a) is not equal to 1.
- The Quadratic Formula: While not strictly a factoring method, the quadratic formula can be used to find the roots of the quadratic equation, which can then be used to determine the factors.
For the expression 2x² + 7x - 49, we will primarily focus on the AC method and trial and error, as these are often the most efficient approaches for this type of quadratic.
Applying the AC Method to 2x² + 7x - 49
The AC method is a powerful technique for factoring quadratic expressions of the form ax² + bx + c. It involves the following steps:
- Identify a, b, and c: In our case, a = 2, b = 7, and c = -49.
- Calculate ac: Multiply a and c: 2 * (-49) = -98.
- Find two numbers that multiply to ac and add up to b: We need to find two numbers that multiply to -98 and add up to 7. This is the crucial step, and it often requires some careful consideration. Let's list the factors of -98:
- 1 and -98
- -1 and 98
- 2 and -49
- -2 and 49
- 7 and -14
- -7 and 14 We can see that -7 and 14 satisfy the conditions: (-7) * 14 = -98 and (-7) + 14 = 7.
- Rewrite the middle term (bx) using the two numbers found: We replace 7x with -7x + 14x, so the expression becomes 2x² - 7x + 14x - 49.
- Factor by grouping: Group the first two terms and the last two terms: (2x² - 7x) + (14x - 49).
- Factor out the greatest common factor (GCF) from each group: From the first group, we can factor out x, and from the second group, we can factor out 7: x(2x - 7) + 7(2x - 7).
- Factor out the common binomial factor: Notice that both terms now have a common factor of (2x - 7). Factoring this out, we get (2x - 7)(x + 7).
Therefore, the factored form of 2x² + 7x - 49 is (2x - 7)(x + 7). This is a key result that demonstrates the power of the AC method in factoring quadratic expressions.
Trial and Error Method for 2x² + 7x - 49
The trial and error method, while sometimes less systematic than the AC method, can be a valuable tool, especially for simpler quadratic expressions or when you have a good intuition about the factors. This method involves making educated guesses about the factors and then checking if they multiply back to the original expression. Let's apply this method to 2x² + 7x - 49.
We know that the factored form will be two binomials of the form (Ax + B)(Cx + D), where A, B, C, and D are constants. To get the quadratic term 2x², we know that A and C must multiply to 2. The possibilities are:
- A = 2, C = 1
- A = 1, C = 2
Similarly, to get the constant term -49, B and D must multiply to -49. The possibilities are:
- B = 7, D = -7
- B = -7, D = 7
- B = 1, D = -49
- B = -1, D = 49
- B = 49, D = -1
- B = -49, D = 1
Now, we need to try different combinations of these possibilities and check if the middle term (7x) is obtained when we expand the binomials.
Let's start with the combination (2x + 7)(x - 7). Expanding this, we get:
2x² - 14x + 7x - 49 = 2x² - 7x - 49
This does not match our original expression, so this combination is incorrect.
Next, let's try (2x - 7)(x + 7). Expanding this, we get:
2x² + 14x - 7x - 49 = 2x² + 7x - 49
This matches our original expression, so this combination is correct. Therefore, the factored form of 2x² + 7x - 49 is (2x - 7)(x + 7). The trial and error method highlights the importance of systematically testing different combinations until the correct factorization is found.
Verifying the Factored Form
After factoring a quadratic expression, it's always a good practice to verify your result. This can be done by expanding the factored form and checking if it matches the original expression. Let's verify our factored form of 2x² + 7x - 49, which is (2x - 7)(x + 7).
Expanding the factored form, we get:
(2x - 7)(x + 7) = 2x(x + 7) - 7(x + 7)
Distribute 2x and -7:
= 2x² + 14x - 7x - 49
Combine like terms:
= 2x² + 7x - 49
The expanded form matches our original expression, so our factorization is correct. This verification step is crucial for ensuring accuracy and building confidence in your factoring skills.
Identifying the Correct Answer
Now that we have successfully factored the quadratic expression 2x² + 7x - 49 and verified our result, we can identify the correct answer from the given options.
- A. (x - 7)(2x - 7)
- B. (2x + 7)(x + 7)
- C. (x + 7)(2x - 7)
- D. (2x + 7)(x - 7)
Our factored form is (2x - 7)(x + 7), which matches option C. (x + 7)(2x - 7). Note that the order of the factors does not matter since multiplication is commutative. For example, (2x - 7)(x + 7) is equivalent to (x + 7)(2x - 7).
Therefore, the correct answer is C. (x + 7)(2x - 7). This step solidifies the entire factoring process, ensuring that the final answer is accurate and aligns with the initial problem.
Conclusion: Mastering Quadratic Factoring
Factoring quadratic expressions is an essential skill in algebra, with applications in various mathematical contexts. In this article, we have provided a detailed guide on how to factor the quadratic expression 2x² + 7x - 49. We explored the AC method and trial and error, demonstrating the systematic steps involved in factoring. We also emphasized the importance of verifying the factored form to ensure accuracy.
By mastering these techniques, you will be well-equipped to tackle a wide range of quadratic factoring problems. Remember to practice regularly and apply these methods to different types of quadratic expressions. With consistent effort, you can develop a strong understanding of factoring and excel in algebra and beyond. Factoring is not just a mathematical skill; it's a problem-solving tool that empowers you to approach complex problems with confidence and precision. The ability to factor quadratic expressions will be invaluable in higher-level math courses and in real-world applications where mathematical modeling is essential.
