Factoring Quadratic Equations: Solve 66x² - 55x = 0
Let's dive into the world of quadratic equations and explore how to solve them by factoring. In this article, we'll tackle the equation 66x² - 55x = 0 step by step. Factoring is a powerful technique that simplifies complex equations, making them easier to solve. So, if you're ready to master this skill, let's get started!
Understanding Quadratic Equations
Before we jump into factoring, let's make sure we're all on the same page about what a quadratic equation is. A quadratic equation is a polynomial equation 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. Understanding this foundational concept is crucial for tackling factoring and other solution methods.
In our case, the equation 66x² - 55x = 0 fits this form. Here, a = 66, b = -55, and c = 0. Recognizing the coefficients is the first step in determining the best approach to solve the equation. While the quadratic formula can solve any quadratic equation, factoring is often quicker and simpler when applicable. In this article, we'll focus specifically on using the factoring method to find the solutions.
Quadratic equations pop up in various real-world scenarios, from physics problems involving projectile motion to engineering calculations for designing structures. Their ability to model parabolic curves makes them invaluable in many fields. For instance, understanding quadratic equations helps engineers calculate the trajectory of a ball thrown in the air or design the curve of a bridge. Therefore, mastering the techniques to solve these equations opens doors to solving numerous practical problems.
Factoring: The Key to Simplifying
Factoring is a technique used to break down a mathematical expression into a product of simpler expressions. Think of it like finding the building blocks of a number or an equation. For example, the number 12 can be factored into 2 × 2 × 3. Similarly, in algebra, we can factor polynomials into simpler polynomials. This skill is particularly useful when dealing with quadratic equations because it allows us to rewrite the equation in a form that's easier to solve. By identifying common factors and rewriting the equation, we simplify the problem and make it more manageable.
When we factor a quadratic equation, we aim to rewrite it as a product of two binomials (expressions with two terms). For instance, the equation x² - 5x + 6 can be factored into (x - 2)(x - 3). This transformation is key because it leverages the Zero Product Property, which we'll discuss shortly. Factoring isn't just about finding any factors; it's about finding the specific factors that help us solve the equation. The goal is to rewrite the quadratic expression as a product of linear expressions, making it easier to find the values of 'x' that satisfy the equation.
Factoring can seem daunting at first, but with practice, it becomes a powerful tool in your mathematical arsenal. It's a fundamental technique not only for solving quadratic equations but also for simplifying algebraic expressions, solving higher-degree polynomials, and even in calculus. By mastering factoring, you'll gain a deeper understanding of the structure of mathematical expressions and become more confident in your problem-solving abilities. The ability to factor efficiently is a cornerstone of algebraic manipulation and a skill that will benefit you in various mathematical contexts.
Step-by-Step Factoring of 66x² - 55x = 0
Now, let's get our hands dirty and factor the equation 66x² - 55x = 0. The first step in any factoring problem is to look for a common factor among the terms. In this case, both 66x² and -55x have a common factor. Identifying this common factor is crucial because it simplifies the equation significantly. Overlooking this step can make the factoring process more complicated, so it's always the first thing you should check.
Looking at the coefficients, 66 and 55, we can see that both are divisible by 11. Also, both terms have 'x' as a factor. So, the greatest common factor (GCF) is 11x. Factoring out 11x from the equation, we get: 11x(6x - 5) = 0. This step transforms the original quadratic expression into a product of two factors, which is the key to solving the equation. By pulling out the GCF, we've made the equation much easier to handle, setting the stage for the next step in finding the solutions.
This factored form, 11x(6x - 5) = 0, is where the magic happens. We've successfully rewritten the quadratic equation as a product of two factors, which allows us to use the Zero Product Property to find the solutions. This step is a perfect example of how factoring simplifies complex equations and makes them more accessible. By identifying and extracting the greatest common factor, we've set the stage for the final step in solving the equation.
Applying the Zero Product Property
The Zero Product Property is a fundamental principle in algebra that states: if the product of two or more factors is zero, then at least one of the factors must be zero. This property is the cornerstone of solving equations by factoring. It allows us to take a factored equation and break it down into simpler equations that we can solve individually. Understanding and applying this property correctly is essential for finding the solutions to factored quadratic equations.
In our case, we have the factored equation 11x(6x - 5) = 0. This means that either 11x = 0 or (6x - 5) = 0 (or both). This is where the Zero Product Property comes into play. We set each factor equal to zero and solve for 'x'. This step transforms one equation into two simpler equations, each of which can be solved independently. By applying the Zero Product Property, we're effectively turning a complex problem into two straightforward ones.
Solving 11x = 0 is quite simple. Divide both sides by 11, and we get x = 0. This is our first solution. For the second factor, 6x - 5 = 0, we need to isolate 'x'. We first add 5 to both sides, giving us 6x = 5. Then, we divide both sides by 6, resulting in x = 5/6. Thus, we have our second solution. The Zero Product Property, therefore, is the bridge that connects factoring to the actual solutions of the equation. It's the critical step that allows us to extract the values of 'x' that make the equation true.
The Solutions: x = 0 and x = 5/6
So, we've successfully factored the equation and applied the Zero Product Property. Our solutions are x = 0 and x = 5/6. These are the two values of 'x' that, when plugged back into the original equation 66x² - 55x = 0, will make the equation true. This is a critical concept to understand: solutions to an equation are the values that satisfy the equation. To verify our solutions, we can substitute each value back into the original equation and check if the left side equals the right side.
Let's check x = 0 first: 66(0)² - 55(0) = 0. This simplifies to 0 - 0 = 0, which is true. So, x = 0 is indeed a solution. Now let's check x = 5/6: 66(5/6)² - 55(5/6) = 0. This looks a bit more complex, but let's break it down. (5/6)² = 25/36. So, we have 66(25/36) - 55(5/6) = 0. Simplifying further, we get (66 * 25) / 36 - (55 * 5) / 6 = 0. This becomes 1650/36 - 275/6 = 0. To subtract these fractions, we need a common denominator, which is 36. So, we rewrite 275/6 as (275 * 6) / 36 = 1650/36. Now we have 1650/36 - 1650/36 = 0, which is true. Therefore, x = 5/6 is also a solution. Verifying solutions is an essential step in problem-solving because it ensures that our calculations are correct and that we've found the accurate answers.
These solutions, x = 0 and x = 5/6, represent the points where the parabola described by the quadratic equation intersects the x-axis. In a graphical representation, these values are the x-intercepts of the parabola. Understanding the relationship between the algebraic solutions and the graphical representation of the equation provides a deeper insight into the nature of quadratic equations. The solutions are not just numbers; they have a geometric interpretation as well.
Conclusion
In this article, we've walked through the process of solving the quadratic equation 66x² - 55x = 0 by factoring. We began by understanding what a quadratic equation is and how it can be identified. Then, we delved into the technique of factoring, emphasizing the importance of finding common factors. By factoring out 11x, we simplified the equation to 11x(6x - 5) = 0. We then applied the Zero Product Property, which allowed us to break the equation into two simpler equations: 11x = 0 and 6x - 5 = 0. Solving these equations gave us the solutions x = 0 and x = 5/6. Finally, we verified these solutions by substituting them back into the original equation, confirming their validity. This step-by-step approach provides a clear and methodical way to tackle quadratic equations using factoring.
Factoring is a valuable skill in algebra, not only for solving quadratic equations but also for simplifying expressions and solving other types of equations. By mastering factoring, you gain a deeper understanding of the structure of mathematical expressions and improve your problem-solving abilities. The ability to factor efficiently is a cornerstone of algebraic manipulation and a skill that will benefit you in various mathematical contexts. Practice is key to becoming proficient in factoring, so make sure to tackle a variety of problems to solidify your understanding.
By understanding the Zero Product Property and how it links factoring to finding solutions, you can confidently approach and solve a wide range of quadratic equations. Remember, each solution represents a specific value of 'x' that makes the equation true, and these values often have significant interpretations in real-world applications. The solutions are not just abstract numbers; they often represent critical points or values in the context of a problem.
For further exploration and practice on quadratic equations, consider visiting Khan Academy's Quadratic Equations Section. You'll find a wealth of resources, including videos, practice problems, and articles, to help you deepen your understanding and skills.
