Factoring: Solve X(x+2) = 3x(x-1) - 3 Equation

Hey guys! Today, we're diving into a fun math problem that involves factoring. We'll break down how to solve the equation x(x+2) = 3x(x-1) - 3 step-by-step. So, if you've ever felt a little lost when it comes to factoring, you're in the right place. Let's jump right in and make math a little less mysterious!

Understanding the Problem

Before we start crunching numbers, let’s make sure we understand exactly what we're dealing with. Our main goal here is to factor and solve the given equation, which is x(x+2) = 3x(x-1) - 3. This looks like a quadratic equation, but to confirm, we need to simplify and rearrange it into the standard form, which is ax² + bx + c = 0. Recognizing the type of equation we're working with helps us choose the right tools and techniques to solve it.

When we talk about factoring, we mean rewriting the equation as a product of its factors. Think of it like reverse multiplication. For example, if we have x² + 5x + 6, factoring it means finding two binomials that multiply together to give us this quadratic. Once we have it factored, we can use the zero-product property to find the solutions. This property states that if the product of two factors is zero, then at least one of the factors must be zero. It’s a handy trick that makes solving factored equations much easier.

So, why is factoring so important? Well, it’s not just a math exercise. Factoring is a fundamental skill in algebra and calculus. It pops up in various applications, from physics to engineering, where we need to find the roots or zeros of a polynomial. Understanding factoring can also help simplify complex equations, making them easier to work with. Plus, it's like unlocking a puzzle – there's a certain satisfaction in finding the right factors!

Step-by-Step Solution

Let's get into the nitty-gritty and solve this equation together! The best way to tackle this is to break it down into manageable steps. Don't worry, we'll take it slow and explain each part along the way. Grab your pen and paper, and let's dive in!

Step 1: Expand Both Sides of the Equation

First things first, we need to get rid of those parentheses. To do this, we'll expand both sides of the equation. On the left side, we have x(x+2), and on the right side, we have 3x(x-1) - 3. Let's use the distributive property (remember that?!) to multiply through.

On the left, x * (x + 2) becomes x² + 2x. Easy peasy! Now, let's tackle the right side. We have 3x * (x - 1), which gives us 3x² - 3x. But don't forget, we still have that -3 hanging out at the end. So, the right side becomes 3x² - 3x - 3. Now our equation looks like this:

x² + 2x = 3x² - 3x - 3

Step 2: Rearrange the Equation to the Standard Quadratic Form

Now we need to get everything on one side to set the equation to zero. This is crucial because it sets us up to factor (or use the quadratic formula, if needed). We want to end up with an equation in the form ax² + bx + c = 0. To do this, let's subtract x² and 2x from both sides of the equation. This will keep everything balanced and help us rearrange the terms.

Subtracting x² from both sides gives us: 2x = 2x² - 3x - 3.

Next, subtract 2x from both sides: 0 = 2x² - 5x - 3.

Now our equation is in the standard quadratic form:

2x² - 5x - 3 = 0

Step 3: Factor the Quadratic Expression

This is the heart of the problem! We need to factor the quadratic expression 2x² - 5x - 3. Factoring means finding two binomials that, when multiplied together, give us this quadratic. There are different methods to factor, but a common one is the trial-and-error method or the AC method.

Let’s use the AC method here. First, we multiply the coefficient of x² (which is 2) by the constant term (which is -3). This gives us -6. Now, we need to find two numbers that multiply to -6 and add up to the coefficient of x (which is -5). Those numbers are -6 and 1.

Now we rewrite the middle term, -5x, using these numbers: 2x² - 6x + x - 3 = 0.

Next, we factor by grouping. We group the first two terms and the last two terms: (2x² - 6x) + (x - 3) = 0.

Now, factor out the greatest common factor (GCF) from each group. From the first group, we can factor out 2x, which gives us 2x(x - 3). From the second group, we can factor out 1, which gives us 1(x - 3). So, our equation becomes:

2x(x - 3) + 1(x - 3) = 0

Notice that both terms now have a common factor of (x - 3). We can factor this out, giving us:

(2x + 1)(x - 3) = 0

Woohoo! We've successfully factored the quadratic expression!

Step 4: Apply the Zero-Product Property

This is where the magic happens! The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have (2x + 1)(x - 3) = 0. This means that either (2x + 1) = 0 or (x - 3) = 0.

Let’s solve each of these equations separately.

First, solve 2x + 1 = 0. Subtract 1 from both sides: 2x = -1. Then, divide by 2: x = -1/2.

Next, solve x - 3 = 0. Add 3 to both sides: x = 3.

So, we have two solutions:

x = -1/2 and x = 3

Step 5: Verify the Solutions

Last but not least, it’s always a good idea to check our answers. Plug each solution back into the original equation to make sure they work. This helps prevent errors and gives us confidence in our results.

Let's check x = -1/2:

Original equation: x(x + 2) = 3x(x - 1) - 3

Substitute x = -1/2: (-1/2)((-1/2) + 2) = 3(-1/2)((-1/2) - 1) - 3

Simplify: (-1/2)(3/2) = (-3/2)(-3/2) - 3

Further simplify: -3/4 = 9/4 - 3

Convert 3 to a fraction with a denominator of 4: -3/4 = 9/4 - 12/4

Simplify: -3/4 = -3/4

It checks out!

Now let's check x = 3:

Original equation: x(x + 2) = 3x(x - 1) - 3

Substitute x = 3: 3(3 + 2) = 3(3)(3 - 1) - 3

Simplify: 3(5) = 9(2) - 3

Further simplify: 15 = 18 - 3

Simplify: 15 = 15

It checks out too!

Both solutions are valid. We've done it!

Common Mistakes to Avoid

Factoring can be tricky, and there are a few common pitfalls that students often stumble into. But don’t worry, we’re here to shine a light on those mistakes so you can steer clear of them!

One frequent error is not distributing correctly. Remember, when you're expanding expressions, make sure you multiply every term inside the parentheses by the term outside. For instance, in our equation, we had 3x(x - 1). Make sure you multiply 3x by both x and -1. Forgetting to multiply by one of the terms can throw off the entire solution.

Another common mistake is messing up the signs. This usually happens when rearranging the equation to the standard form or when factoring. Always double-check your signs, especially when moving terms across the equals sign or when determining the factors of a quadratic. A small sign error can lead to completely different results.

Not setting the equation to zero before factoring is another big no-no. The zero-product property only works when one side of the equation is zero. So, before you start factoring, make sure you've rearranged the equation into the ax² + bx + c = 0 form.

Finally, forgetting to check your solutions is a missed opportunity to catch errors. Always plug your solutions back into the original equation to verify they work. It’s like a final safety check that can save you from incorrect answers.

Tips and Tricks for Factoring

Now that we’ve conquered a challenging equation, let’s talk about some tips and tricks that can make factoring a breeze. These strategies will help you become a factoring pro in no time!

First, always look for a greatest common factor (GCF). Before diving into more complex factoring methods, see if there’s a common factor you can pull out of all the terms. Factoring out the GCF first can simplify the expression and make the rest of the factoring process much easier.

Practice makes perfect! The more you practice factoring, the better you’ll become at recognizing patterns and quickly finding factors. Try working through a variety of problems, from simple quadratics to more complex expressions. Over time, you’ll develop an intuition for factoring that will serve you well in math and beyond.

Use factoring shortcuts when possible. For example, learn to recognize the difference of squares (a² - b²) and perfect square trinomials (a² + 2ab + b² or a² - 2ab + b²). These patterns can be factored quickly using specific formulas, saving you time and effort.

Don’t be afraid to use the AC method. This method is especially helpful for factoring quadratics with a leading coefficient not equal to 1. It provides a systematic approach to finding the factors and can be a lifesaver when dealing with more complex expressions.

If you’re stuck, remember that there are other methods to solve quadratic equations, like the quadratic formula. If factoring isn’t working, don’t hesitate to switch gears and try a different approach. The goal is to find the solutions, and sometimes that means using a different tool in your math toolkit.

Real-World Applications of Factoring

Okay, so we've mastered the mechanics of factoring, but you might be wondering,