Solving For X In The Equation 4(3x² + 2) = 776
Introduction: A Mathematical Quest to Find the Value of x
In the realm of mathematics, equations serve as puzzles waiting to be solved, and the equation 4(3x² + 2) = 776 is no exception. This equation presents us with a fascinating challenge: to determine the value of the unknown variable, x, that satisfies the given relationship. This article embarks on a detailed exploration of the equation, employing algebraic techniques and step-by-step guidance to unravel the mystery of x. We'll delve into the heart of the equation, dissecting its components, and systematically applying mathematical principles to isolate x and reveal its hidden value. Whether you're a seasoned mathematician or a curious student, this journey promises to enhance your understanding of algebraic problem-solving and the elegance of mathematical reasoning.
Setting the Stage: Understanding the Equation
Before we dive into the solution, let's first grasp the essence of the equation 4(3x² + 2) = 776. This equation is a quadratic equation, characterized by the presence of the term x². The equation involves several mathematical operations: multiplication, addition, and exponentiation. Our goal is to isolate x on one side of the equation, which will involve reversing these operations in a strategic manner. The equation's structure hints at a methodical approach, where we'll gradually peel away the layers surrounding x until its true value is revealed. Understanding the equation's anatomy is crucial for navigating the solution process effectively.
The Journey Begins: Step-by-Step Solution
Now, let's embark on our step-by-step journey to solve the equation 4(3x² + 2) = 776. Each step is a carefully orchestrated maneuver, bringing us closer to the ultimate goal of unveiling x. We will present the equation solving process in a clear, logical and detailed way so that the readers can understand easily. Let's dive into the solving process:
Step 1: Distribute the 4
The first step in simplifying the equation is to distribute the 4 across the terms inside the parentheses. This involves multiplying 4 by both 3x² and 2. This distribution helps us to eliminate the parentheses and make the equation more manageable. Mathematically, this step can be represented as:
4 * (3x² + 2) = 4 * 3x² + 4 * 2 = 12x² + 8
Thus, our equation now transforms into:
12x² + 8 = 776
This step lays the foundation for further simplification, paving the way for isolating the x² term.
Step 2: Isolate the Term with x²
Our next objective is to isolate the term containing x², which is 12x². To achieve this, we need to eliminate the constant term, 8, from the left side of the equation. This can be accomplished by subtracting 8 from both sides of the equation. Subtracting the same value from both sides maintains the balance of the equation, ensuring that the equality remains valid. This step is crucial for bringing us closer to isolating x. The mathematical representation of this step is:
12x² + 8 - 8 = 776 - 8
Simplifying this, we get:
12x² = 768
Now, the term with x² stands alone on the left side, ready for the next step in our solution.
Step 3: Divide to Isolate x²
To further isolate x², we need to eliminate the coefficient 12. This can be done by dividing both sides of the equation by 12. Dividing both sides by the same non-zero value preserves the equation's balance and helps us to isolate x². This step brings us even closer to finding the value of x. The mathematical representation of this step is:
(12x²) / 12 = 768 / 12
Simplifying this, we obtain:
x² = 64
Now, x² is completely isolated, revealing its value as 64. This is a significant milestone in our journey to solve for x.
Step 4: Take the Square Root
The final step in solving for x is to take the square root of both sides of the equation. This operation undoes the squaring, allowing us to determine the value(s) of x. Remember that taking the square root can result in both positive and negative solutions, as both the positive and negative square roots, when squared, yield the same positive value. This is a critical point to consider for complete accuracy. Mathematically, this step can be represented as:
√(x²) = ±√64
Simplifying this, we get:
x = ±8
Therefore, the solutions for x are 8 and -8.
The Value of x: Unveiled!
After our step-by-step journey through the equation 4(3x² + 2) = 776, we have successfully unveiled the value of x. The solutions are x = 8 and x = -8. These values, when substituted back into the original equation, will satisfy the equality, confirming their validity. Our quest to find x has reached its triumphant conclusion, showcasing the power of algebraic manipulation and the beauty of mathematical solutions.
Verification: Ensuring the Accuracy of Our Solutions
To ensure the accuracy of our solutions, we must substitute the values we obtained for x back into the original equation. This process of verification serves as a crucial check, confirming that our solutions indeed satisfy the given relationship. Let's embark on this verification journey, substituting both x = 8 and x = -8 into the original equation 4(3x² + 2) = 776.
Verifying x = 8
Substituting x = 8 into the equation, we get:
4(3(8)² + 2) = 776
Let's simplify this step by step:
4(3(64) + 2) = 776
4(192 + 2) = 776
4(194) = 776
776 = 776
As we can see, the left-hand side of the equation equals the right-hand side when x = 8. This confirms that x = 8 is indeed a valid solution.
Verifying x = -8
Now, let's substitute x = -8 into the equation:
4(3(-8)² + 2) = 776
Simplifying this step by step:
4(3(64) + 2) = 776
4(192 + 2) = 776
4(194) = 776
776 = 776
Again, the left-hand side of the equation equals the right-hand side, this time when x = -8. This confirms that x = -8 is also a valid solution.
The Verdict: Solutions Confirmed
Through our rigorous verification process, we have confirmed that both x = 8 and x = -8 are indeed valid solutions to the equation 4(3x² + 2) = 776. This verification step underscores the importance of checking our work, ensuring the accuracy and reliability of our mathematical solutions. Our quest has not only led us to the solutions but also provided us with the confidence that these solutions are correct.
Conclusion: The Power of Algebraic Problem-Solving
In this article, we have embarked on a mathematical journey to solve the equation 4(3x² + 2) = 776. Through a series of carefully executed algebraic steps, we successfully isolated x and discovered its values: x = 8 and x = -8. We then verified these solutions, solidifying our confidence in their accuracy. This exploration has showcased the power of algebraic problem-solving, demonstrating how we can unravel complex equations by systematically applying mathematical principles.
The process of solving this equation has highlighted several key concepts in algebra, including distribution, isolating variables, and the importance of considering both positive and negative roots when dealing with square roots. These concepts are fundamental to mathematical understanding and problem-solving. The ability to manipulate equations and isolate variables is a valuable skill that extends beyond the realm of mathematics, finding applications in various fields, including science, engineering, and finance.
Moreover, this journey has emphasized the importance of verification in mathematics. By substituting our solutions back into the original equation, we ensured their validity and gained confidence in our results. This practice of verification is a cornerstone of mathematical rigor, encouraging accuracy and precision in problem-solving.
In conclusion, solving the equation 4(3x² + 2) = 776 has been more than just a mathematical exercise; it has been a testament to the power of algebraic reasoning and the importance of methodical problem-solving. The values of x have been unveiled, and the journey has enriched our understanding of mathematical principles and their applications.
