What Are The Values Of X That Satisfy The Equation $3x^2 + 17x - 6 = 0$?

Introduction to Quadratic Equations

In this comprehensive article, we will delve into the step-by-step solution of the quadratic equation $3x^2 + 17x - 6 = 0$. Quadratic equations, a cornerstone of algebra, are polynomial equations of the second degree. They take the general form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we aim to solve for. The solutions to a quadratic equation, also known as roots or zeros, represent the values of $x$ that satisfy the equation. Mastering the techniques for solving quadratic equations is crucial for various applications in mathematics, physics, engineering, and computer science. From modeling projectile motion to designing electrical circuits, quadratic equations play a pivotal role in describing and predicting real-world phenomena. This article aims to provide a clear and detailed explanation of one common method for solving quadratic equations: factoring. By understanding the underlying principles and following the steps outlined below, you'll gain confidence in tackling a wide range of quadratic equations. So, let's embark on this mathematical journey and unravel the solution to $3x^2 + 17x - 6 = 0$. Remember, practice is key to mastering any mathematical concept, and quadratic equations are no exception. By working through various examples and applying the techniques learned, you'll develop a strong foundation in algebra and enhance your problem-solving skills.

Problem Statement: $3x^2 + 17x - 6 = 0$

Our primary objective is to find the values of $x$ that satisfy the given quadratic equation: $3x^2 + 17x - 6 = 0$. This equation is a standard quadratic equation, characterized by its second-degree term ($3x^2$), a linear term ($17x$), and a constant term ($-6$). To solve this equation, we will employ the factoring method, a technique that involves breaking down the quadratic expression into a product of two binomial expressions. Factoring is a powerful tool for solving quadratic equations, especially when the roots are rational numbers. It allows us to rewrite the equation in a form that makes it easier to identify the values of $x$ that make the equation true. Before diving into the factoring process, it's essential to understand the underlying principles. We are essentially looking for two numbers that, when multiplied, give us the product of the leading coefficient (3) and the constant term (-6), which is -18, and when added, give us the middle coefficient (17). This process might seem daunting at first, but with practice, it becomes a straightforward method for solving quadratic equations. The ability to factor quadratic expressions is a fundamental skill in algebra, and it opens doors to solving more complex mathematical problems. In the following sections, we will break down the factoring process step-by-step, ensuring a clear understanding of each stage. By mastering this technique, you'll be well-equipped to solve a wide array of quadratic equations and further your mathematical prowess.

Step-by-Step Solution

To solve the quadratic equation $3x^2 + 17x - 6 = 0$, we will use the factoring method. This method involves rewriting the middle term ($17x$) as a sum of two terms, such that the resulting expression can be factored by grouping. Here's a detailed breakdown of the steps:

Step 1: Rewrite the Middle Term

The key to factoring this quadratic equation lies in finding two numbers that multiply to the product of the leading coefficient (3) and the constant term (-6), which is -18, and add up to the middle coefficient (17). These two numbers are 18 and -1. We can rewrite the middle term, $17x$, as the sum of $18x$ and $-x$. This step is crucial because it allows us to split the quadratic expression into four terms, which can then be factored by grouping. The process of finding these two numbers might require some trial and error, but with practice, it becomes more intuitive. The goal is to transform the original quadratic expression into a form that is conducive to factoring. By rewriting the middle term, we set the stage for the next step, which involves grouping the terms and extracting common factors. This is a fundamental technique in factoring quadratic expressions and is widely applicable to various algebraic problems. The ability to identify the correct numbers for rewriting the middle term is a cornerstone of mastering the factoring method.

$$3x^2 + 17x - 6 = 3x^2 + 18x - x - 6$$

Step 2: Factor by Grouping

Now that we've rewritten the middle term, we have four terms: $3x^2 + 18x - x - 6$. The next step is to factor by grouping. This involves grouping the first two terms and the last two terms together and then factoring out the greatest common factor (GCF) from each group. From the first group, $3x^2 + 18x$, the GCF is $3x$. Factoring out $3x$, we get $3x(x + 6)$. From the second group, $-x - 6$, the GCF is $-1$. Factoring out $-1$, we get $-1(x + 6)$. Notice that both groups now share a common binomial factor, $(x + 6)$. This is a crucial observation, as it allows us to factor the entire expression. Factoring by grouping is a powerful technique that simplifies the expression and brings us closer to finding the solutions. This step demonstrates the elegance of factoring, where we transform a complex expression into a more manageable form. The ability to identify and extract common factors is a fundamental skill in algebra, and it is essential for solving various types of equations and inequalities. The next step will involve factoring out the common binomial factor, which will lead us to the final factored form of the quadratic expression.

$$3x(x + 6) - 1(x + 6)$$

Step 3: Factor out the Common Binomial

In the previous step, we arrived at the expression $3x(x + 6) - 1(x + 6)$. We can see that the binomial $(x + 6)$ is a common factor in both terms. To factor out the common binomial, we treat $(x + 6)$ as a single entity and factor it out from the entire expression. This is a key step in the factoring process, as it simplifies the expression further and reveals the factors of the quadratic equation. When we factor out $(x + 6)$, we are left with $(3x - 1)$ as the other factor. This results in the factored form of the quadratic expression: $(3x - 1)(x + 6)$. The factored form is crucial because it allows us to easily identify the roots of the equation. Each factor represents a potential solution to the equation. By setting each factor equal to zero, we can solve for the values of $x$ that make the equation true. This step highlights the power of factoring in solving quadratic equations. It transforms the equation into a form that is easily solvable, providing a clear path to finding the roots. The ability to recognize and factor out common binomials is a valuable skill in algebra and is widely applicable in various mathematical contexts.

$$(3x - 1)(x + 6) = 0$$

Step 4: Set Each Factor to Zero

Now that we have the factored form of the equation, $(3x - 1)(x + 6) = 0$, we can use the zero-product property. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is the cornerstone of solving factored equations. It allows us to break down the original equation into two simpler equations, each of which can be solved independently. To apply this property, we set each factor equal to zero: $3x - 1 = 0$ and $x + 6 = 0$. This step is crucial because it transforms the single quadratic equation into two linear equations, which are much easier to solve. By setting each factor to zero, we create a direct pathway to finding the roots of the original equation. This step demonstrates the elegance of the factoring method, where a complex problem is broken down into simpler, more manageable parts. The zero-product property is a fundamental concept in algebra and is essential for solving various types of equations, including polynomial equations and rational equations.

$$3x - 1 = 0 ext{ or } x + 6 = 0$$

Step 5: Solve for x

We now have two simple linear equations to solve: $3x - 1 = 0$ and $x + 6 = 0$. Solving for $x$ in each equation will give us the solutions to the original quadratic equation. For the first equation, $3x - 1 = 0$, we add 1 to both sides to get $3x = 1$, and then divide both sides by 3 to get x = rac{1}{3}. For the second equation, $x + 6 = 0$, we subtract 6 from both sides to get $x = -6$. These two values of $x$ are the roots of the quadratic equation. They represent the points where the parabola defined by the equation intersects the x-axis. Solving for $x$ in these linear equations is a straightforward process, but it is a crucial step in finding the solutions to the quadratic equation. This step demonstrates the importance of basic algebraic manipulations in solving more complex problems. By isolating $x$ in each equation, we arrive at the final solutions, which are the values of $x$ that satisfy the original equation. These solutions can be used to analyze the behavior of the quadratic function and to solve related problems in various fields, such as physics and engineering.

For $3x - 1 = 0$:

$$3x = 1$$

$$x = \frac{1}{3}$$

For $x + 6 = 0$:

$$x = -6$$

Solutions

Based on the step-by-step solution, the values of $x$ that satisfy the quadratic equation $3x^2 + 17x - 6 = 0$ are $x = \frac{1}{3}$ and $x = -6$. These are the roots of the equation, representing the points where the quadratic function intersects the x-axis. The solutions are obtained by factoring the quadratic expression and applying the zero-product property. Each solution represents a value of $x$ that, when substituted back into the original equation, will make the equation true. These solutions are not just numerical answers; they provide valuable information about the behavior of the quadratic function. For instance, they can be used to determine the axis of symmetry, the vertex, and the overall shape of the parabola. The solutions also have practical applications in various fields, such as physics and engineering, where quadratic equations are used to model real-world phenomena. Therefore, understanding how to find and interpret the solutions of quadratic equations is a fundamental skill in mathematics. In summary, the solutions $x = \frac{1}{3}$ and $x = -6$ are the definitive answers to the problem, and they provide a complete understanding of the equation's roots.

Conclusion

In this article, we have thoroughly explored the process of solving the quadratic equation $3x^2 + 17x - 6 = 0$ using the factoring method. We began by rewriting the middle term, $17x$, as the sum of $18x$ and $-x$, which allowed us to factor the expression by grouping. We then factored out the common binomial factor, $(x + 6)$, and applied the zero-product property to set each factor equal to zero. Finally, we solved the resulting linear equations to find the solutions: $x = \frac{1}{3}$ and $x = -6$. This step-by-step approach provides a clear and concise method for solving quadratic equations by factoring. The factoring method is a powerful tool for solving quadratic equations, especially when the roots are rational numbers. It involves breaking down the quadratic expression into a product of two binomial expressions, which makes it easier to identify the solutions. Mastering this technique requires practice and a solid understanding of algebraic principles. However, with consistent effort, it becomes a valuable skill for solving various mathematical problems. The solutions to a quadratic equation are not just numerical answers; they provide insights into the behavior of the quadratic function and have practical applications in various fields. Therefore, understanding the factoring method and its applications is essential for anyone pursuing mathematics, science, or engineering.

Further Practice

To solidify your understanding of solving quadratic equations by factoring, it's crucial to engage in further practice. Try solving similar quadratic equations using the same method. You can also explore other methods for solving quadratic equations, such as the quadratic formula and completing the square. Comparing and contrasting these methods will enhance your understanding of quadratic equations and their solutions. Practice problems can be found in textbooks, online resources, and worksheets. Working through a variety of problems will help you develop your problem-solving skills and gain confidence in your ability to solve quadratic equations. Additionally, consider exploring real-world applications of quadratic equations. This will help you see the relevance of the concepts you are learning and make the learning process more engaging. Quadratic equations are used in various fields, such as physics, engineering, and finance, to model and solve real-world problems. By practicing and exploring these applications, you will not only master the techniques for solving quadratic equations but also develop a deeper appreciation for their importance and versatility. Remember, the key to success in mathematics is consistent practice and a willingness to explore new concepts. So, keep practicing and keep learning!