Factoring Quadratic Equations 3x² + 26x + 35 A Step-by-Step Guide

Hey guys! Have you ever felt like you're staring at a quadratic equation and it's staring right back at you, all mysterious and intimidating? Well, fret no more! Today, we're going to break down how to factor the quadratic equation 3x² + 26x + 35. Trust me, it's not as scary as it looks. We’ll dive deep into the method of factoring, making sure you not only understand how to do it but also why it works. So, grab your pencils, and let’s get started on this mathematical adventure!

Understanding Quadratic Equations

Before we jump into factoring, let’s make sure we're all on the same page about what a quadratic equation actually is. Quadratic equations, at their core, are polynomial equations of the second degree. This means the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, the term x² would vanish, and we'd be left with a linear equation instead.

Now, why are quadratic equations so important? Well, they pop up everywhere in the real world! Think about the trajectory of a ball thrown in the air – that's a parabola, which is described by a quadratic equation. Or consider the design of a bridge, the optimal shape for a satellite dish, or even the growth of a population under certain conditions. Quadratic equations are fundamental in physics, engineering, economics, and many other fields. Understanding them opens up a whole new world of problem-solving possibilities.

In our specific equation, 3x² + 26x + 35, we can identify the coefficients as follows: a = 3, b = 26, and c = 35. These coefficients are the key to unlocking the factored form of the equation. The goal of factoring is to rewrite the quadratic equation as a product of two binomials. In other words, we want to find two expressions of the form (px + q) and (rx + s) such that (px + q)(rx + s) = 3x² + 26x + 35. Sounds like a puzzle, right? That's because it is! And we're about to solve it.

Factoring is a crucial skill in algebra because it allows us to find the solutions (also called roots or zeros) of the equation. The solutions are the values of 'x' that make the equation equal to zero. Once we have the factored form, we can easily find these solutions by setting each factor equal to zero and solving for 'x'. This is based on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This principle is the foundation for solving many algebraic equations, not just quadratics. By mastering factoring, you gain a powerful tool for tackling more complex mathematical problems in the future. So, let's dive deeper into the factoring process itself!

The Factoring Method: A Step-by-Step Guide

Alright, let's get down to the nitty-gritty of factoring our equation, 3x² + 26x + 35. The method we'll be using is often called the 'ac method' or the 'grouping method.' It's a systematic way to break down the quadratic expression and rewrite it in a form that's easier to factor.

Step 1: Multiply 'a' and 'c'

The first thing we need to do is multiply the coefficients 'a' and 'c'. In our equation, a = 3 and c = 35. So, we calculate 3 * 35 = 105. This number, 105, is going to be crucial in the next step. It's like the magic number that unlocks the factoring puzzle. This initial step transforms the trinomial factoring problem into a search for two numbers with specific properties, simplifying the overall process.

Step 2: Find Two Numbers

Now comes the tricky part, but don't worry, we'll get through it together. We need to find two numbers that multiply to 105 (the result from Step 1) and add up to 26 (the coefficient 'b' in our equation). This is where a little trial and error might come into play, but there are some strategies we can use to make it easier. Start by listing the factors of 105: 1 and 105, 3 and 35, 5 and 21, 7 and 15. Now, look at these pairs and see which one adds up to 26. Aha! It's 5 and 21. These are our magic numbers! This step is pivotal as it decomposes the middle term into two terms that facilitate grouping, making the subsequent factorization straightforward. The careful selection of these two numbers is the cornerstone of the 'ac' method.

Step 3: Rewrite the Middle Term

With our magic numbers in hand, we can rewrite the middle term (26x) using 5 and 21. So, 26x becomes 5x + 21x. Our equation now looks like this: 3x² + 5x + 21x + 35. Notice that we haven't changed the value of the equation; we've simply rewritten it in a way that will allow us to factor by grouping. This rewriting is a strategic move that sets the stage for factoring by grouping, a technique that relies on identifying common factors within pairs of terms.

Step 4: Factor by Grouping

This is where the 'grouping' part of the method comes into play. We're going to group the first two terms and the last two terms together: (3x² + 5x) + (21x + 35). Now, we'll factor out the greatest common factor (GCF) from each group. From the first group, 3x² + 5x, the GCF is x. Factoring out x, we get x(3x + 5). From the second group, 21x + 35, the GCF is 7. Factoring out 7, we get 7(3x + 5). So, our equation now looks like this: x(3x + 5) + 7(3x + 5). The beauty of this step is that it reveals a common binomial factor, which is the key to the final factorization.

Step 5: Factor Out the Common Binomial

Look closely at our equation: x(3x + 5) + 7(3x + 5). Do you see the common binomial factor? It's (3x + 5)! We can factor this out just like we factored out the GCF in the previous step. When we factor out (3x + 5), we're left with (x + 7). So, the factored form of our equation is (3x + 5)(x + 7). Congratulations! We've successfully factored the quadratic equation! This final step consolidates the previous efforts by extracting the common binomial factor, resulting in the factored form of the quadratic equation. This factorization not only simplifies the equation but also provides direct access to its solutions.

Verifying the Solution

Before we pat ourselves on the back too hard, it's always a good idea to verify our solution. How do we do that? By expanding the factored form and making sure it matches the original equation. Let's expand (3x + 5)(x + 7) using the FOIL method (First, Outer, Inner, Last):

• First: 3x * x = 3x²
• Outer: 3x * 7 = 21x
• Inner: 5 * x = 5x
• Last: 5 * 7 = 35

Now, let's add those terms together: 3x² + 21x + 5x + 35. Combining like terms, we get 3x² + 26x + 35. Ta-da! It's the same as our original equation. This confirms that our factoring is correct. Verifying the solution is a critical step in the problem-solving process. It not only validates the factoring but also reinforces understanding and builds confidence. By expanding the factored form and comparing it to the original equation, we ensure accuracy and prevent errors.

Tips and Tricks for Factoring

Factoring can sometimes feel like a puzzle, but with practice, it becomes much easier. Here are a few tips and tricks to help you along the way:

1. Practice, practice, practice: The more you factor, the better you'll become at recognizing patterns and applying the methods. Work through various examples, starting with simpler ones and gradually moving to more complex problems.
2. Look for common factors first: Before diving into the 'ac' method, always check if there's a common factor that can be factored out from all the terms. This can simplify the equation and make it easier to factor further. For example, if you had 6x² + 52x + 70, you could factor out a 2 first, resulting in 2(3x² + 26x + 35), and then factor the quadratic expression inside the parentheses.
3. Use the signs to your advantage: Pay attention to the signs of the coefficients. If 'c' is positive, the two numbers you're looking for will have the same sign (either both positive or both negative). If 'c' is negative, the two numbers will have opposite signs. If 'b' is positive, the larger number will be positive; if 'b' is negative, the larger number will be negative. These sign relationships can significantly narrow down the possibilities and make the search for the correct numbers more efficient.
4. Don't be afraid to try and err: Factoring often involves some trial and error. If your first guess doesn't work, don't get discouraged. Just try a different pair of numbers. Sometimes, writing out the factors of 'ac' can help you visualize the possibilities and spot the correct pair more easily. The key is to be systematic and persistent in your approach.
5. Check your work: As we discussed earlier, always verify your solution by expanding the factored form. This is the best way to catch any mistakes and ensure that you've factored correctly. It's a small investment of time that can save you from errors in later steps of a problem.

Common Mistakes to Avoid

Factoring can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Forgetting to factor out the GCF: As mentioned earlier, always look for a greatest common factor first. Forgetting to do this can lead to more complicated factoring later on and may even make it impossible to factor correctly.
• Incorrectly identifying the numbers that multiply to 'ac' and add up to 'b': This is the most crucial step in the 'ac' method, and a mistake here will throw off the entire factoring process. Double-check your calculations and make sure you've found the correct pair of numbers.
• Making sign errors: Sign errors are very common in factoring. Pay close attention to the signs of the coefficients and the signs of the numbers you're using to rewrite the middle term. A simple sign error can completely change the factored form.
• Factoring the expression incompletely: Make sure you've factored the expression as much as possible. Sometimes, after factoring by grouping, you may need to factor out a common factor from the resulting binomials.
• Not checking your work: We can't stress this enough: always verify your solution by expanding the factored form. This is the best way to catch any mistakes and ensure that you've factored correctly. Skipping this step can lead to incorrect solutions and a lack of confidence in your factoring abilities.

Conclusion

So there you have it, guys! We've successfully factored the quadratic equation 3x² + 26x + 35. Remember, the key to mastering factoring is understanding the method, practicing regularly, and being mindful of the common mistakes. With enough practice, you'll be able to factor quadratic equations with confidence and ease. Keep practicing, and soon you'll be a factoring pro! Happy solving!