Solve Quadratics By Factoring: A Step-by-Step Guide

Hey guys! Let's dive into the world of quadratic equations and explore a powerful method for solving them: factoring. Quadratic equations are those polynomial equations with a highest degree of 2, meaning they have a term with $x^2$. They pop up everywhere in math and real-world applications, so mastering how to solve them is super important. In this guide, we'll break down the factoring method step-by-step, using a specific example to make things crystal clear. We'll tackle the equation $x^2 - 84 = 2x - 4$. So, buckle up, and let's get started!

Understanding Quadratic Equations

Before we jump into the nitty-gritty of factoring, let's make sure we're all on the same page about what quadratic equations actually are. A quadratic equation is generally expressed in the standard form: $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The 'x' represents the variable we're trying to solve for. These equations can have up to two solutions, also known as roots or zeros. Understanding the structure of a quadratic equation is the first step in conquering it.

Now, you might be wondering, why is 'a' not allowed to be zero? Well, if 'a' were zero, the $x^2$ term would disappear, and we'd be left with a linear equation ($bx + c = 0$) instead of a quadratic one. So, that $x^2$ term is what makes a quadratic equation, well, quadratic! Another key thing to remember is that the solutions to a quadratic equation are the values of 'x' that make the equation true. These solutions can be real numbers, complex numbers, or a combination of both. The nature of the solutions depends on the coefficients 'a', 'b', and 'c', which we'll explore further as we delve into different solving methods. Knowing the standard form helps us identify the coefficients easily, which is crucial for applying various techniques, including factoring. So, keep that $ax^2 + bx + c = 0$ form in your mind as we move forward!

The Power of Factoring

Factoring is a technique used to simplify expressions or equations by breaking them down into smaller, multiplicative parts. When it comes to solving quadratic equations, factoring helps us rewrite the equation in a form where we can easily identify the solutions. The basic idea behind factoring is to express the quadratic expression as a product of two binomials. Think of it like this: we're trying to undo the multiplication that would have created the quadratic expression in the first place.

Why is factoring so powerful? Well, it relies on a fundamental property of real numbers called the zero-product property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In mathematical terms, if A * B = 0, then either A = 0 or B = 0 (or both). This simple yet profound principle is the cornerstone of solving quadratic equations by factoring. Once we've factored the quadratic expression into two binomials, we can set each binomial equal to zero and solve for 'x'. This gives us the two possible solutions to the equation. Factoring isn't just a mathematical trick; it's a systematic way of finding the roots of a quadratic equation by leveraging the zero-product property. It’s like unlocking a secret code that reveals the hidden solutions. So, with a solid understanding of this property, we’re well-equipped to tackle the factoring process itself.

Step-by-Step: Solving $x^2 - 84 = 2x - 4$ by Factoring

Okay, let's get our hands dirty and solve the equation $x^2 - 84 = 2x - 4$ using factoring. We'll break it down into manageable steps so you can follow along easily. Trust me, it's not as scary as it looks!

Step 1: Rearrange the Equation into Standard Form

The first thing we need to do is get our equation into the standard quadratic form: $ax^2 + bx + c = 0$. This means we need to move all the terms to one side of the equation, leaving zero on the other side. To do this, we'll subtract $2x$ and add $4$ to both sides of the equation:

$x^2 - 84 - 2x + 4 = 2x - 4 - 2x + 4$

Simplifying this, we get:

$x^2 - 2x - 80 = 0$

Now, our equation is in the standard form, with $a = 1$, $b = -2$, and $c = -80$. Having the equation in this format makes it much easier to identify the coefficients and proceed with the factoring process. This step is crucial because it sets the stage for applying the factoring techniques we'll discuss next. Without this initial rearrangement, it would be much harder to see the structure of the quadratic expression and determine how to factor it. So, always remember to get your equation into standard form before moving on to the next steps!

Step 2: Factor the Quadratic Expression

Now comes the fun part: factoring! We need to find two numbers that multiply to give us 'c' (-80) and add up to give us 'b' (-2). This might sound like a puzzle, but with a little practice, it becomes second nature. Let's think about the factors of -80. We're looking for a pair where one number is positive and the other is negative (since their product is negative), and their difference is 2.

After a bit of mental math (or maybe a quick jotting down of factor pairs), we'll find that the numbers are -10 and 8. Why? Because (-10) * 8 = -80, and (-10) + 8 = -2. Bingo!

Now we can rewrite our quadratic expression in factored form:

$(x - 10)(x + 8) = 0$

This factored form is equivalent to our original quadratic expression, but it's now expressed as a product of two binomials. This is the key step in solving by factoring, as it allows us to use the zero-product property. Factoring is like unlocking a hidden structure within the equation, revealing the roots in a more accessible way. There are other factoring techniques too, but for this particular equation, finding the right number pairs is the most straightforward approach. So, practice identifying these pairs, and you'll become a factoring pro in no time!

Step 3: Apply the Zero-Product Property

Remember the zero-product property we talked about earlier? This is where it comes into play. 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:

$(x - 10)(x + 8) = 0$

This means either $(x - 10) = 0$ or $(x + 8) = 0$ (or both). Now we have two simple linear equations to solve!

This step is the heart of the factoring method, as it directly connects the factored form to the solutions of the equation. By applying the zero-product property, we transform a single quadratic equation into two simpler linear equations, which are much easier to solve. It's like breaking a complex problem into smaller, more manageable pieces. The elegance of this step lies in its simplicity and effectiveness. It allows us to isolate the variable 'x' and find the values that make the original equation true. So, appreciate the power of the zero-product property – it's the bridge between factoring and finding solutions!

Step 4: Solve for x

Let's solve each of these linear equations separately.

For $(x - 10) = 0$, we add 10 to both sides:

$x = 10$

For $(x + 8) = 0$, we subtract 8 from both sides:

$x = -8$

So, we have two solutions: $x = 10$ and $x = -8$. These are the values of 'x' that make the original equation $x^2 - 84 = 2x - 4$ true. We've successfully found the roots of the quadratic equation by factoring!

This final step is where all our previous work comes to fruition. By solving the linear equations, we unveil the solutions that satisfy the original quadratic equation. Each solution represents a point where the parabola (the graph of a quadratic equation) intersects the x-axis. Finding these points is a fundamental goal in solving quadratic equations, as they provide key insights into the behavior of the quadratic function. So, with $x = 10$ and $x = -8$, we've not only solved the equation but also gained a deeper understanding of its roots. High five! You've mastered solving a quadratic equation by factoring.

Checking Our Solutions

It's always a good idea to check our solutions to make sure they're correct. To do this, we'll substitute each value of 'x' back into the original equation and see if it holds true.

Checking $x = 10$

Substitute $x = 10$ into $x^2 - 84 = 2x - 4$:

$(10)^2 - 84 = 2(10) - 4$

$100 - 84 = 20 - 4$

$16 = 16$ Yep, it checks out!

Checking $x = -8$

Substitute $x = -8$ into $x^2 - 84 = 2x - 4$:

$(-8)^2 - 84 = 2(-8) - 4$

$64 - 84 = -16 - 4$

$-20 = -20$ Woohoo! This one checks out too!

Since both solutions satisfy the original equation, we can confidently say that $x = 10$ and $x = -8$ are the correct solutions. Checking our work is a crucial step in problem-solving, as it helps us catch any potential errors and ensure the accuracy of our results. It's like the final seal of approval on our solution. So, always take the time to check your answers – it's well worth the effort!

Conclusion

Alright, guys! We've successfully navigated the world of quadratic equations and learned how to solve them by factoring. We took the equation $x^2 - 84 = 2x - 4$, rearranged it into standard form, factored the quadratic expression, applied the zero-product property, and solved for 'x'. We even checked our solutions to make sure they were correct. You're now equipped with a powerful tool for tackling quadratic equations!

Factoring is a fundamental technique in algebra, and mastering it will open doors to more advanced mathematical concepts. Remember, practice makes perfect, so keep working on those factoring skills. The more you practice, the more comfortable and confident you'll become. And who knows, maybe you'll even start to enjoy the puzzle-solving aspect of factoring! So, keep up the great work, and remember that math can be fun and rewarding. You've got this!