Solutions To 3(x-4)(x+5)=0 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article provides a detailed exploration of how to solve the equation 3(x-4)(x+5)=0. We will discuss the underlying principles, step-by-step solutions, and the importance of understanding the zero-product property. Whether you're a student grappling with algebra or simply curious about mathematical problem-solving, this guide will equip you with the knowledge and confidence to tackle similar equations.
Understanding the Zero-Product Property
The zero-product property is the cornerstone of solving equations in the form of factored expressions. 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 zero. Mathematically, if A * B = 0, then either A = 0, B = 0, or both A and B are equal to zero. This seemingly simple principle is incredibly powerful and allows us to break down complex equations into manageable parts. For example, consider the equation (x - 2)(x + 3) = 0. According to the zero-product property, either (x - 2) must equal zero or (x + 3) must equal zero. This leads to two separate equations: x - 2 = 0 and x + 3 = 0. Solving these individual equations gives us the solutions x = 2 and x = -3. Understanding and applying the zero-product property correctly is essential for finding the correct solutions to factored equations.
When faced with an equation like 3(x - 4)(x + 5) = 0, the zero-product property becomes our primary tool. The equation is already conveniently factored, which means we can directly apply the property. The equation presents three factors: 3, (x - 4), and (x + 5). According to the zero-product property, at least one of these factors must be equal to zero for the entire product to be zero. The constant factor 3, however, can never be zero. This leaves us with the factors (x - 4) and (x + 5). Thus, we set each of these factors equal to zero and solve for x. This process involves creating two separate equations: x - 4 = 0 and x + 5 = 0. By solving each equation independently, we can determine the values of x that satisfy the original equation. The application of the zero-product property not only simplifies the problem but also provides a clear and logical pathway to finding the solutions.
By utilizing the zero-product property effectively, we can systematically solve factored equations and gain a deeper understanding of algebraic principles. This method is not only applicable to simple quadratic equations but also extends to more complex polynomial equations. The key is to identify the factors, set each factor to zero, and then solve the resulting equations. Mastering this technique is crucial for success in algebra and beyond. In the following sections, we will apply this property specifically to the equation 3(x - 4)(x + 5) = 0, providing a step-by-step solution to illustrate the process.
Step-by-Step Solution of 3(x-4)(x+5)=0
To solve the equation 3(x-4)(x+5)=0, we will methodically apply the zero-product property. This involves identifying the factors, setting each factor to zero, and then solving the resulting equations. This step-by-step approach ensures clarity and accuracy in finding the solutions. The solutions we find will be the values of x that make the equation true. Equations like this are essential in various mathematical and real-world applications, including physics, engineering, and economics, where determining the roots of a function is a common task. The process we follow here provides a solid foundation for solving more complex equations in the future.
Step 1: Identify the Factors
The equation 3(x - 4)(x + 5) = 0 presents three factors: 3, (x - 4), and (x + 5). These are the individual components that, when multiplied together, result in zero. Recognizing these factors is the first crucial step in applying the zero-product property. The constant factor, 3, is a straightforward number, while the expressions (x - 4) and (x + 5) are binomial factors involving the variable x. These binomial factors represent linear expressions, meaning the highest power of x in each factor is 1. Identifying these factors correctly sets the stage for the next steps in the solution process. Understanding the structure of the equation and the nature of its factors is essential for choosing the correct method to solve it.
Step 2: Apply the Zero-Product Property
According to the zero-product property, if the product of several factors is zero, then at least one of the factors must be zero. In this case, since 3(x - 4)(x + 5) = 0, either 3 = 0, (x - 4) = 0, or (x + 5) = 0. However, we know that 3 cannot equal zero. Therefore, we only need to consider the cases where (x - 4) = 0 or (x + 5) = 0. This step simplifies the problem into two separate, more manageable equations. By setting each variable factor to zero, we create a pathway to isolate x and find its possible values. The zero-product property is a powerful tool because it transforms a single, potentially complex equation into multiple simpler equations that can be solved individually. This method is widely used in algebra and forms the basis for solving many types of equations.
Step 3: Solve the Equations
We now have two separate equations to solve:
- x - 4 = 0
- x + 5 = 0
For the first equation, x - 4 = 0, we add 4 to both sides to isolate x:
x - 4 + 4 = 0 + 4
x = 4
For the second equation, x + 5 = 0, we subtract 5 from both sides to isolate x:
x + 5 - 5 = 0 - 5
x = -5
Thus, we have found two solutions for x: x = 4 and x = -5. These values are the roots of the equation, meaning they are the values that make the equation true. Solving each equation involves using basic algebraic manipulations to isolate the variable. Adding or subtracting the same value from both sides of the equation maintains the equality and allows us to progressively simplify the equation until we find the value of x. This process is fundamental to solving algebraic equations and demonstrates the importance of understanding inverse operations. Once we have found the solutions, it's a good practice to check them by substituting them back into the original equation to ensure they satisfy the equation.
Verification of the Solutions
After finding the solutions to an equation, it's crucial to verify their correctness. This step ensures that the values we found indeed satisfy the original equation and that no errors were made during the solution process. Verification involves substituting each solution back into the original equation and checking if the equation holds true. This practice is essential for building confidence in our answers and for identifying any potential mistakes. In the context of the equation 3(x-4)(x+5)=0, we will substitute x = 4 and x = -5 separately to confirm their validity.
Verifying x = 4
To verify x = 4, we substitute this value into the original equation:
3(x - 4)(x + 5) = 0
3(4 - 4)(4 + 5) = 0
3(0)(9) = 0
0 = 0
Since the equation holds true when x = 4, we can confirm that x = 4 is a valid solution. The substitution process involves replacing the variable x with the numerical value 4 and then simplifying the expression step by step. The key is to follow the order of operations (PEMDAS/BODMAS) to ensure accurate simplification. In this case, the factor (4 - 4) becomes zero, which makes the entire product zero, satisfying the equation. This verification step provides us with confidence that our solution is correct and that we have applied the principles of algebra accurately.
Verifying x = -5
Next, we verify x = -5 by substituting it into the original equation:
3(x - 4)(x + 5) = 0
3(-5 - 4)(-5 + 5) = 0
3(-9)(0) = 0
0 = 0
Again, the equation holds true when x = -5, confirming that it is also a valid solution. Similar to the previous verification, we replace x with -5 and simplify the expression. The factor (-5 + 5) becomes zero, which, when multiplied by any other factors, results in zero. This satisfies the equation and confirms that x = -5 is a correct solution. By verifying both solutions, we have demonstrated that they both make the original equation true, and we can confidently state that these are the correct solutions to the equation 3(x-4)(x+5)=0. Verification is a critical step in mathematical problem-solving, as it ensures accuracy and understanding.
Conclusion: The Solutions and Their Significance
In conclusion, by applying the zero-product property and performing the necessary algebraic steps, we have determined that the solutions to the equation 3(x-4)(x+5)=0 are x = 4 and x = -5. These values are the roots of the equation, meaning they are the values of x that make the equation true. Understanding how to find these solutions is a fundamental skill in algebra and has broad applications in various fields.
The significance of these solutions extends beyond just solving a mathematical equation. In graphical terms, the solutions represent the x-intercepts of the quadratic function defined by the equation y = 3(x-4)(x+5). The x-intercepts are the points where the graph of the function crosses the x-axis, and they provide valuable information about the function's behavior. Furthermore, in real-world contexts, these solutions could represent critical points in a model, such as the points where a profit function equals zero or the times at which a projectile hits the ground. The ability to find these solutions is therefore essential for problem-solving in many practical situations.
The process of solving the equation also highlights the importance of the zero-product property. This property allows us to break down a complex equation into simpler parts, making it easier to solve. Without this property, solving factored equations would be significantly more challenging. The zero-product property is a cornerstone of algebra and is used extensively in solving various types of equations, including polynomial equations of higher degrees. Mastering this property is crucial for success in algebra and beyond.
Moreover, the step-by-step approach we used to solve the equation demonstrates a systematic method for problem-solving. By identifying the factors, applying the zero-product property, solving the resulting equations, and verifying the solutions, we have followed a clear and logical process. This systematic approach can be applied to many other mathematical problems and is a valuable skill to develop. It emphasizes the importance of breaking down complex problems into smaller, more manageable steps and of checking our work to ensure accuracy.
In summary, the solutions x = 4 and x = -5 to the equation 3(x-4)(x+5)=0 are not just numerical answers; they are critical values that provide insights into the equation and its applications. The process of finding these solutions underscores the importance of fundamental algebraic principles and problem-solving strategies. By understanding these concepts, you can confidently tackle similar equations and apply them to real-world scenarios. This detailed exploration aims to equip you with the knowledge and skills necessary for success in algebra and related fields.
