Solving (3x + 1)(x + 2) = 0 A Step-by-Step Guide
Understanding Quadratic Equations and how to solve them is a fundamental concept in algebra. This article delves into the process of solving the quadratic equation (3x + 1)(x + 2) = 0, providing a step-by-step explanation to enhance your understanding. We'll cover the key principles, the zero-product property, and the practical application of these concepts. Mastering this method is crucial for tackling more complex algebraic problems and real-world applications.
The foundation of solving this equation lies in recognizing that it's a quadratic equation in factored form. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, 'x') is 2. These equations often take the general form ax² + bx + c = 0, where a, b, and c are constants. However, the equation we're dealing with, (3x + 1)(x + 2) = 0, is presented in a factored form, which simplifies the solving process considerably. Factored form means that the equation is expressed as a product of two or more factors, each containing the variable 'x'. This form is particularly useful because it directly leads to the solutions using the zero-product property. Understanding the structure of quadratic equations, whether in standard or factored form, is the initial step towards finding the values of 'x' that satisfy the equation. By recognizing the equation's form, we can strategically apply the appropriate techniques to arrive at the solution efficiently.
The Zero-Product Property
The zero-product property is the cornerstone of solving equations in factored form. This property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In simpler terms, if A * B = 0, then either A = 0 or B = 0 (or both). This property is incredibly powerful because it allows us to break down a complex equation into simpler ones. In the context of our equation, (3x + 1)(x + 2) = 0, the zero-product property dictates that either (3x + 1) must equal zero or (x + 2) must equal zero. This transforms our single quadratic equation into two linear equations, which are much easier to solve individually. The beauty of this property lies in its ability to convert a multiplication problem into a set of simpler equations, each of which can be solved independently. By understanding and applying the zero-product property, we can effectively find the values of 'x' that make the original equation true. This property is not only applicable to quadratic equations but also extends to any equation where factors are multiplied to equal zero, making it a versatile tool in algebra.
Applying the Zero-Product Property to (3x + 1)(x + 2) = 0
To effectively solve the equation (3x + 1)(x + 2) = 0, we apply the zero-product property by setting each factor equal to zero. This leads us to two separate equations: 3x + 1 = 0 and x + 2 = 0. Each of these equations is a linear equation, which can be solved by isolating 'x'. The first equation, 3x + 1 = 0, requires us to first subtract 1 from both sides, resulting in 3x = -1. Then, we divide both sides by 3 to solve for 'x', giving us x = -1/3. The second equation, x + 2 = 0, is even simpler to solve. We subtract 2 from both sides to isolate 'x', which gives us x = -2. Thus, by applying the zero-product property, we've transformed the original quadratic equation into two straightforward linear equations, each yielding a solution for 'x'. These solutions represent the values of 'x' that, when substituted back into the original equation, will make the equation true. This step-by-step process demonstrates the practical application of the zero-product property and its importance in solving factored equations.
Solving the Linear Equations
Now, let's thoroughly examine the process of solving the two linear equations we obtained from applying the zero-product property: 3x + 1 = 0 and x + 2 = 0. Solving linear equations involves isolating the variable 'x' on one side of the equation. For the first equation, 3x + 1 = 0, our goal is to get 'x' by itself. We begin by subtracting 1 from both sides of the equation to eliminate the constant term on the left side. This gives us 3x = -1. Next, to isolate 'x', we divide both sides of the equation by 3, the coefficient of 'x'. This results in x = -1/3, which is one of the solutions to our original quadratic equation. For the second equation, x + 2 = 0, the process is even more straightforward. To isolate 'x', we simply subtract 2 from both sides of the equation. This gives us x = -2, the second solution to the quadratic equation. These steps demonstrate the fundamental techniques for solving linear equations, which are essential building blocks for tackling more complex algebraic problems. By mastering these techniques, we can confidently solve for variables in a wide range of equations.
Solution 1: 3x + 1 = 0
To methodically solve the equation 3x + 1 = 0, we follow a series of algebraic steps aimed at isolating the variable 'x'. The first step involves removing the constant term from the left side of the equation. We achieve this by subtracting 1 from both sides of the equation, ensuring that the equation remains balanced. This operation yields the new equation 3x = -1. Now, 'x' is still attached to the coefficient 3, meaning it's being multiplied by 3. To isolate 'x', we need to perform the inverse operation, which is division. We divide both sides of the equation by 3, which cancels out the multiplication on the left side. This leaves us with x = -1/3. This is the first solution to our quadratic equation. It represents a value of 'x' that, when substituted back into the original equation (3x + 1)(x + 2) = 0, will make the equation true. This process highlights the importance of inverse operations in solving algebraic equations. By carefully applying these operations, we can systematically isolate the variable and find its value.
Solution 2: x + 2 = 0
The equation x + 2 = 0 presents a straightforward scenario for solving for 'x'. In this case, 'x' is being added to 2, and our objective is to isolate 'x' on one side of the equation. To accomplish this, we perform the inverse operation of addition, which is subtraction. We subtract 2 from both sides of the equation, ensuring that the equation remains balanced and the equality holds. This operation effectively cancels out the +2 on the left side, leaving us with x = -2. This is the second solution to our quadratic equation. It indicates another value of 'x' that satisfies the original equation (3x + 1)(x + 2) = 0. Substituting x = -2 back into the original equation will confirm that the equation holds true. This simple yet effective method demonstrates the power of using inverse operations to isolate variables and solve equations. It's a fundamental technique in algebra that is widely applicable across various types of equations.
Verifying the Solutions
Verifying the solutions is a crucial step in the problem-solving process, ensuring the accuracy of our results. To verify our solutions, x = -1/3 and x = -2, we substitute each value back into the original equation, (3x + 1)(x + 2) = 0, and check if the equation holds true. Let's start with x = -1/3. Substituting this value into the equation, we get (3*(-1/3) + 1)((-1/3) + 2) = 0. Simplifying the expression inside the first parenthesis, 3*(-1/3) equals -1, so we have (-1 + 1) in the first parenthesis. This simplifies to 0. In the second parenthesis, (-1/3) + 2 equals 5/3. So, the equation becomes 0 * (5/3) = 0, which is indeed true. This confirms that x = -1/3 is a valid solution. Now, let's verify x = -2. Substituting this value into the equation, we get (3*(-2) + 1)((-2) + 2) = 0. Simplifying the expression inside the first parenthesis, 3*(-2) equals -6, so we have (-6 + 1), which simplifies to -5. In the second parenthesis, (-2) + 2 equals 0. So, the equation becomes -5 * 0 = 0, which is also true. This confirms that x = -2 is a valid solution. By substituting both solutions back into the original equation, we have demonstrated that both values satisfy the equation, providing a high level of confidence in our answers.
Substituting x = -1/3 into (3x + 1)(x + 2) = 0
The process of substituting x = -1/3 into the original equation (3x + 1)(x + 2) = 0 is a meticulous verification step. By replacing 'x' with '-1/3', we aim to confirm whether this value indeed makes the equation true. Let's break down the substitution and simplification. First, we replace each instance of 'x' with '-1/3', resulting in (3*(-1/3) + 1)((-1/3) + 2) = 0. Next, we simplify the expression within each set of parentheses. Inside the first parentheses, 3 multiplied by -1/3 equals -1, so we have (-1 + 1). Inside the second parentheses, we add -1/3 and 2. To do this, we need a common denominator, so we rewrite 2 as 6/3. Thus, we have (-1/3 + 6/3), which equals 5/3. Now, our equation looks like this: (0)(5/3) = 0. Since any number multiplied by 0 is 0, the left side of the equation simplifies to 0. Therefore, we have 0 = 0, which is a true statement. This verification confirms that x = -1/3 is indeed a valid solution to the equation (3x + 1)(x + 2) = 0. This step-by-step substitution demonstrates the accuracy of our solution and reinforces the principles of algebraic manipulation.
Substituting x = -2 into (3x + 1)(x + 2) = 0
Verifying the solution x = -2 involves substituting this value into the original equation (3x + 1)(x + 2) = 0 and checking if the equation holds true. We begin by replacing each 'x' in the equation with '-2', which gives us (3*(-2) + 1)((-2) + 2) = 0. Now, we simplify each set of parentheses separately. Inside the first parentheses, we have 3 multiplied by -2, which equals -6. Adding 1 to -6 gives us -5. Inside the second parentheses, we have -2 plus 2, which equals 0. So, our equation now looks like this: (-5)(0) = 0. Since any number multiplied by 0 is 0, the left side of the equation simplifies to 0. This leaves us with 0 = 0, which is a true statement. This result confirms that x = -2 is a valid solution to the original equation. This verification process demonstrates the importance of checking solutions to ensure accuracy and reinforces our understanding of how solutions satisfy equations.
Conclusion
In conclusion, solving the quadratic equation (3x + 1)(x + 2) = 0 involves several key steps, each playing a crucial role in arriving at the correct solutions. We began by understanding the structure of the equation and recognizing that it was presented in factored form, which is particularly advantageous for solving. The application of the zero-product property was the next pivotal step, allowing us to break down the original equation into two simpler linear equations: 3x + 1 = 0 and x + 2 = 0. We then systematically solved each of these linear equations, isolating 'x' to find the solutions x = -1/3 and x = -2. To ensure the accuracy of our solutions, we performed a verification step, substituting each value back into the original equation and confirming that the equation held true. This process not only validated our answers but also reinforced our understanding of how solutions satisfy equations. By following these steps, we have successfully solved the quadratic equation and gained valuable insights into the principles of algebra. This comprehensive approach to problem-solving, from understanding the problem to verifying the solutions, is essential for mastering mathematical concepts and applying them effectively.