Calculate Y = 6(2x^2 + 5x - 3) When X = -2
Introduction
In this mathematics article, we will delve into the process of calculating the value of y in the equation y = 6(2x2 + 5x - 3)** when x = -2. This type of problem is a fundamental concept in algebra and is crucial for understanding more complex mathematical concepts. We will break down the steps involved, providing a clear and comprehensive explanation to help readers grasp the methodology. This exercise not only reinforces basic algebraic skills but also enhances problem-solving abilities. By understanding how to substitute values into equations and simplify expressions, you'll be better equipped to tackle a wide range of mathematical challenges. Let's begin by exploring the equation and understanding the role of each component.
The equation y = 6(2x2 + 5x - 3)** is a quadratic equation, where the variable x is squared. The equation states that y is equal to six times the expression inside the parentheses, which includes a quadratic term (2x2), a linear term (5x), and a constant term (-3). To find the value of y when x = -2, we need to substitute -2 for x in the equation and simplify the expression. This involves following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding the order of operations is crucial for accurately solving algebraic equations. First, we'll address the exponent, then multiplication, and finally addition and subtraction within the parentheses before multiplying the entire expression by 6.
Substitution is a fundamental concept in algebra and is used extensively in solving equations and evaluating expressions. When we substitute a value for a variable, we are essentially replacing the variable with that specific value. In this case, we are substituting x with -2. This process allows us to transform a general equation into a specific numerical calculation. It's important to perform the substitution accurately to ensure the final result is correct. The next step involves simplifying the expression after the substitution. This often requires applying the order of operations and combining like terms. Let's proceed step-by-step to ensure clarity and accuracy in our calculation. By the end of this article, you will have a solid understanding of how to solve this type of problem and similar algebraic equations.
Step-by-Step Solution
To calculate the value of y, we first substitute x = -2 into the equation y = 6(2x2 + 5x - 3)**. This substitution is a critical step in solving the problem, as it replaces the variable x with a specific numerical value, allowing us to perform the necessary calculations. Accurate substitution is essential to ensure that we arrive at the correct answer. We must carefully replace each instance of x with -2, paying close attention to the placement of parentheses and the order of operations. Once the substitution is complete, we will have a numerical expression that we can simplify using the rules of arithmetic and algebra.
After substituting, our equation becomes y = 6(2(-2)2 + 5(-2) - 3). Now, we need to simplify the expression inside the parentheses following the order of operations (PEMDAS). This means we first address the exponent, then the multiplication, and finally the addition and subtraction. It's crucial to follow this order to ensure the correct result. Exponents take precedence because they represent repeated multiplication, and addressing them first simplifies the expression significantly. Next, we perform the multiplication operations to further reduce the complexity of the expression. Only after completing these steps can we proceed with the addition and subtraction, which will give us a single numerical value inside the parentheses.
First, we evaluate (-2)2, which equals 4. Our equation now looks like y = 6(2(4) + 5(-2) - 3). Next, we perform the multiplication operations within the parentheses. We have 2 multiplied by 4, which equals 8, and 5 multiplied by -2, which equals -10. So the equation becomes y = 6(8 - 10 - 3). Now, we perform the addition and subtraction from left to right. First, 8 minus 10 equals -2, and then -2 minus 3 equals -5. Therefore, the expression inside the parentheses simplifies to -5. We now have y = 6(-5).
Finally, we multiply 6 by -5, which gives us y = -30. This is the final value of y when x = -2. It's essential to double-check each step to ensure no errors were made during the substitution and simplification process. The negative sign in the result is a direct consequence of multiplying a positive number (6) by a negative number (-5). Understanding how positive and negative numbers interact in multiplication and division is crucial for accurate algebraic calculations. This result indicates the specific value of y for the given value of x, and it represents a point on the graph of the quadratic equation. This step-by-step process allows us to systematically solve the equation and find the desired value.
Summary of Steps
To summarize the steps we took to find the value of y, let's reiterate each action performed. First, we substituted x = -2 into the original equation y = 6(2x2 + 5x - 3)**. This substitution is the cornerstone of solving the problem, allowing us to transform the equation from a general form to a specific calculation. It's crucial to ensure accuracy during this step, as any error here will propagate through the rest of the solution. The substitution effectively replaces the variable x with the numerical value -2, setting the stage for the subsequent simplification.
Second, we simplified the expression inside the parentheses by following the order of operations (PEMDAS). This involved first evaluating the exponent (-2)2, which resulted in 4. Next, we performed the multiplication operations: 2 multiplied by 4 equals 8, and 5 multiplied by -2 equals -10. The expression then became 6(8 - 10 - 3). This step is crucial for reducing the complexity of the equation and isolating the numerical value inside the parentheses. Adhering to the correct order of operations ensures that the calculations are performed in the proper sequence, leading to an accurate result.
Third, we performed the addition and subtraction operations inside the parentheses. Starting from left to right, 8 minus 10 equals -2, and -2 minus 3 equals -5. This simplified the expression inside the parentheses to -5. This stage involves basic arithmetic operations but is a crucial step in arriving at the final value. Accurate addition and subtraction are essential for maintaining the integrity of the calculation. The result, -5, represents the value of the expression inside the parentheses for the given value of x.
Finally, we multiplied 6 by -5, which gave us y = -30. This final multiplication completes the calculation and provides the value of y for x = -2. The result, -30, is the solution to the problem and represents a specific point on the graph of the equation. This summary highlights the importance of each step in the process, from initial substitution to final calculation. By following these steps meticulously, we can confidently solve similar algebraic problems.
Conclusion
In conclusion, by substituting x = -2 into the equation y = 6(2x2 + 5x - 3)** and following the order of operations, we found that y = -30. This exercise demonstrates the importance of understanding and applying algebraic principles to solve equations. The process involved substitution, simplification, and careful adherence to the order of operations (PEMDAS). Each step played a crucial role in arriving at the correct solution. This type of problem is fundamental to algebra and provides a foundation for more advanced mathematical concepts.
Understanding how to substitute values into equations and simplify expressions is a vital skill in mathematics. It's not only essential for solving algebraic problems but also for various applications in science, engineering, and other fields. The ability to accurately perform these calculations allows for the modeling and analysis of real-world phenomena. For instance, in physics, we might use similar equations to describe the motion of objects, and in economics, we might use them to model financial trends. Therefore, mastering these skills opens doors to a wide range of possibilities.
This problem serves as a practical example of how algebraic equations can be solved systematically. The step-by-step approach we followed can be applied to a variety of similar problems. By breaking down the problem into manageable steps, we can ensure accuracy and clarity in our calculations. The process reinforces the importance of attention to detail and the careful application of mathematical rules. Moreover, it highlights the interconnectedness of mathematical concepts, as the order of operations, substitution, and simplification all work together to produce the final result.
In summary, solving the equation y = 6(2x2 + 5x - 3)** for x = -2 required a combination of algebraic skills and attention to detail. The result, y = -30, is a testament to the power of mathematical methods in finding precise solutions. By mastering these fundamental concepts, you can confidently tackle more complex mathematical challenges and apply them to real-world situations. This exercise not only provides a specific answer but also reinforces the general principles of algebraic problem-solving.
