Factoring $7x^2 + 6x - 16$ A Step-by-Step Guide

Factoring quadratic expressions is a fundamental skill in algebra, essential for solving equations, simplifying expressions, and understanding the behavior of polynomial functions. In this comprehensive guide, we will delve into the process of factoring the quadratic expression $7x^2 + 6x - 16$. We will explore different techniques, provide step-by-step explanations, and offer insights into the underlying principles. By the end of this article, you will have a solid understanding of how to factor this specific expression and quadratic expressions in general.

Understanding Quadratic Expressions

Before we dive into the factoring process, it's crucial to understand what a quadratic expression is. A quadratic expression is a polynomial expression of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. The term $ax^2$ is the quadratic term, $bx$ is the linear term, and $c$ is the constant term. In our case, the expression $7x^2 + 6x - 16$ is a quadratic expression with $a = 7$, $b = 6$, and $c = -16$.

Factoring a quadratic expression involves rewriting it as a product of two linear expressions. This process is the reverse of expanding two binomials using the distributive property (also known as the FOIL method). Factoring is a powerful tool because it allows us to find the roots (or zeros) of the quadratic equation, which are the values of $x$ that make the expression equal to zero. These roots correspond to the x-intercepts of the parabola represented by the quadratic expression.

Factoring quadratic expressions is a crucial skill in algebra, and mastering it opens doors to more advanced topics in mathematics. By understanding the structure of quadratic expressions and the techniques for factoring them, you will be well-equipped to tackle a wide range of algebraic problems.

The Factoring Process: A Step-by-Step Approach

To factor the quadratic expression $7x^2 + 6x - 16$, we will use a method that involves finding two numbers that satisfy specific conditions related to the coefficients of the expression. This method is often referred to as the "ac method" or the "grouping method."

Step 1: Identify the Coefficients

First, we need to identify the coefficients $a$, $b$, and $c$ in the quadratic expression $7x^2 + 6x - 16$. As we mentioned earlier, $a = 7$, $b = 6$, and $c = -16$. These coefficients will play a crucial role in our factoring process.

Step 2: Calculate ac

Next, we calculate the product of $a$ and $c$, which is $ac$. In this case, $ac = 7 imes (-16) = -112$. This value is essential because we will use it to find two numbers that meet certain criteria.

Step 3: Find Two Numbers

Now, we need to find two numbers that multiply to $ac$ (which is -112) and add up to $b$ (which is 6). This is often the most challenging part of the factoring process, as it may involve some trial and error. However, there are strategies we can use to make this step easier.

One way to approach this is to list the factors of -112 and check their sums. Since the product is negative, we know that one number must be positive, and the other must be negative. Here are some factor pairs of -112:

• 1 and -112 (sum = -111)
• -1 and 112 (sum = 111)
• 2 and -56 (sum = -54)
• -2 and 56 (sum = 54)
• 4 and -28 (sum = -24)
• -4 and 28 (sum = 24)
• 7 and -16 (sum = -9)
• -7 and 16 (sum = 9)
• 8 and -14 (sum = -6)
• -8 and 14 (sum = 6)

We can see that the pair -8 and 14 satisfies our conditions: $(-8) imes 14 = -112$ and $(-8) + 14 = 6$. These are the two numbers we need.

Step 4: Rewrite the Middle Term

Now that we have found the two numbers, -8 and 14, we rewrite the middle term ($6x$) in our quadratic expression as the sum of two terms using these numbers. So, we rewrite $6x$ as $-8x + 14x$. This gives us:

$7x^2 - 8x + 14x - 16$

Step 5: Factor by Grouping

Next, we factor by grouping. We group the first two terms and the last two terms together:

$(7x^2 - 8x) + (14x - 16)$

Now, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is $x$, and from the second group, the GCF is 2. Factoring out the GCFs, we get:

$x(7x - 8) + 2(7x - 8)$

Notice that both terms now have a common factor of $(7x - 8)$. We can factor this out as well:

$(7x - 8)(x + 2)$

This is the factored form of the quadratic expression $7x^2 + 6x - 16$.

The Solution: $(7x - 8)(x + 2)$

Therefore, the factored form of the expression $7x^2 + 6x - 16$ is $(7x - 8)(x + 2)$. We have successfully factored the quadratic expression using the ac method and factoring by grouping. This solution tells us that the quadratic expression is equivalent to the product of two linear expressions, which can be useful for solving equations and analyzing the behavior of the corresponding quadratic function.

To verify our solution, we can expand the factored form using the distributive property (FOIL method) and check if it matches the original expression:

$(7x - 8)(x + 2) = 7x(x) + 7x(2) - 8(x) - 8(2) = 7x^2 + 14x - 8x - 16 = 7x^2 + 6x - 16$

Since the expanded form matches the original expression, we can be confident that our factoring is correct.

Alternative Methods for Factoring

While the ac method is a versatile technique for factoring quadratic expressions, there are other methods you can use depending on the specific expression. Here are a couple of alternative approaches:

Trial and Error

For simpler quadratic expressions, you can sometimes factor by trial and error. This involves guessing the factors and checking if they multiply to the original expression. While this method can be quick for some expressions, it may not be efficient for more complex quadratics.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, even those that are difficult or impossible to factor using traditional methods. The quadratic formula is given by:

x = rac{-b eq ext{√}(b^2 - 4ac)}{2a}

Once you find the roots, you can use them to construct the factors of the quadratic expression. For example, if the roots are $r_1$ and $r_2$, then the factored form of the expression is $a(x - r_1)(x - r_2)$.

Key Takeaways and Conclusion

Factoring quadratic expressions is a fundamental skill in algebra with numerous applications. In this article, we have explored the process of factoring the expression $7x^2 + 6x - 16$ using the ac method and factoring by grouping. We have also discussed alternative methods for factoring, such as trial and error and the quadratic formula.

Here are some key takeaways from this guide:

• Understanding Quadratic Expressions: A quadratic expression is of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
• The Factoring Process: Factoring involves rewriting a quadratic expression as a product of two linear expressions.
• The ac Method: This method involves finding two numbers that multiply to $ac$ and add up to $b$.
• Factoring by Grouping: After rewriting the middle term, group the terms and factor out the greatest common factor.
• Alternative Methods: Trial and error and the quadratic formula can also be used for factoring.

By mastering the techniques presented in this guide, you will be well-equipped to factor a wide range of quadratic expressions and apply this skill to solve equations and analyze quadratic functions. Factoring is a cornerstone of algebra, and with practice, you can become proficient in this essential skill.

Remember, practice is key to mastering factoring. Work through various examples, try different methods, and don't be afraid to make mistakes. Each mistake is an opportunity to learn and improve your understanding. With consistent effort, you will become a confident and skilled factorer of quadratic expressions.

In conclusion, the factored form of the expression $7x^2 + 6x - 16$ is $(7x - 8)(x + 2)$. We hope this comprehensive guide has provided you with a clear understanding of the factoring process and the techniques involved. Keep practicing, and you'll become a factoring pro in no time!