Solving (x+3)(x-3)=16-2x^2 A Step-by-Step Guide
This article delves into the intricacies of solving the algebraic equation (x+3)(x-3)=16-2x^2. This equation, a fascinating blend of algebraic expressions, challenges us to unravel the mystery of its solutions. Understanding how to solve such equations is crucial in mathematics, especially when dealing with quadratic equations. The ability to manipulate and simplify equations is a cornerstone of mathematical proficiency. In this exploration, we will dissect the equation step-by-step, shedding light on each transformation and the underlying mathematical principles. Our journey will not only provide a solution but will also empower you with the knowledge to tackle similar mathematical challenges with confidence and precision. This equation serves as a perfect example to illustrate various algebraic techniques and concepts. From expanding products to rearranging terms and applying the quadratic formula, every step will be meticulously explained, ensuring that you grasp the ‘how’ and ‘why’ behind the solution. Let’s embark on this mathematical adventure and discover the hidden values of x that satisfy this intriguing equation. By mastering such problems, you enhance your problem-solving skills and deepen your appreciation for the elegance and power of mathematics. Remember, each equation is a puzzle waiting to be solved, and with the right approach, the solution is always within reach.
Understanding the Problem
Before diving into the solution, let's dissect the problem. The equation (x+3)(x-3)=16-2x^2 presents a classic scenario in algebra where we need to find the values of x that satisfy the equation. This particular equation involves both a product of binomials on one side and a quadratic expression on the other, making it a comprehensive exercise in algebraic manipulation. The left side, (x+3)(x-3), immediately hints at the difference of squares, a fundamental concept in algebra. Recognizing this pattern is the first step towards simplifying the equation. On the right side, 16-2x^2 presents a quadratic expression that needs to be carefully considered. The presence of both constant and quadratic terms suggests that we might end up with a quadratic equation, which can then be solved using various methods, such as factoring, completing the square, or applying the quadratic formula. Understanding the structure of the equation is paramount. It guides our approach and ensures that we choose the most efficient path to the solution. Each term and operation in the equation holds a piece of the puzzle, and it is our job to fit these pieces together to reveal the complete picture. By breaking down the equation into its components and understanding the relationships between them, we set the stage for a successful solution. This initial analysis is not just about identifying the type of equation; it's about strategizing our approach and anticipating the challenges that lie ahead. It’s about transforming a seemingly complex problem into a series of manageable steps. So, let’s delve deeper into the anatomy of this equation and prepare ourselves for the mathematical journey ahead.
Step-by-Step Solution
Step 1: Expanding the Left Side
The first step in solving the equation (x+3)(x-3)=16-2x^2 involves expanding the product on the left side. This is where our knowledge of algebraic identities comes into play. The expression (x+3)(x-3) is a classic example of the difference of squares. This identity states that (a+b)(a-b) = a^2 - b^2. Applying this identity to our equation, where a = x and b = 3, we get:
x^2 - 9
This simplification transforms the equation into a more manageable form. By recognizing and applying the difference of squares identity, we've eliminated the need for a lengthy multiplication process. This step highlights the importance of recognizing patterns in algebra, as it can significantly streamline the solution process. Expanding the product is not just about removing parentheses; it's about revealing the underlying structure of the equation and setting the stage for further simplification. The result, x^2 - 9, is a much simpler expression that we can now work with more easily. This step is a testament to the power of algebraic identities in simplifying complex expressions and paving the way for a clearer solution.
Step 2: Rearranging the Equation
Now that we've expanded the left side of the equation, our next task is to rearrange the terms to bring all the variables and constants to one side. This is a crucial step in solving any algebraic equation, as it allows us to isolate the variable and find its value. Our equation now looks like this:
x^2 - 9 = 16 - 2x^2
To rearrange, we want to get all terms involving x on one side and the constants on the other. Let's add 2x^2 to both sides of the equation. This will eliminate the -2x^2 term on the right side:
x^2 - 9 + 2x^2 = 16 - 2x^2 + 2x^2
This simplifies to:
3x^2 - 9 = 16
Next, we want to isolate the term with x^2. To do this, we add 9 to both sides of the equation:
3x^2 - 9 + 9 = 16 + 9
This simplifies to:
3x^2 = 25
By rearranging the equation in this way, we've transformed it into a form that is much easier to solve. We've successfully isolated the x^2 term, which is a significant step towards finding the value of x. This process of rearrangement is a fundamental technique in algebra, and mastering it is essential for solving a wide range of equations. Each step in the rearrangement is carefully chosen to move us closer to the solution, and by following these steps methodically, we can unravel even the most complex equations.
Step 3: Isolating x^2
With the equation rearranged to 3x^2 = 25, our next goal is to isolate x^2. This is a straightforward step that involves dividing both sides of the equation by the coefficient of x^2, which in this case is 3. Dividing both sides by 3, we get:
3x^2 / 3 = 25 / 3
This simplifies to:
x^2 = 25/3
Isolating x^2 is a crucial step because it brings us closer to finding the value of x. Once we have x^2 by itself, we can take the square root of both sides to solve for x. This step is a testament to the power of algebraic manipulation in simplifying equations. By performing a simple division, we've transformed the equation into a form that is much easier to handle. This isolation process is a common technique in algebra, and it's essential for solving equations of all types. It allows us to peel away the layers of the equation and reveal the underlying value of the variable. So, with x^2 now isolated, we're on the verge of discovering the solutions to our equation.
Step 4: Solving for x
Now that we have isolated x^2 in the equation x^2 = 25/3, the final step is to solve for x. To do this, we take the square root of both sides of the equation. It's crucial to remember that when taking the square root, we must consider both the positive and negative roots, as both will satisfy the equation. Taking the square root of both sides, we get:
√x^2 = ±√(25/3)
This simplifies to:
x = ±√(25/3)
We can further simplify this by taking the square root of 25, which is 5:
x = ±5/√3
To rationalize the denominator, we multiply both the numerator and the denominator by √3:
x = ±(5√3) / 3
Therefore, the solutions for x are:
x = (5√3) / 3 and x = -(5√3) / 3
These are the two values of x that satisfy the original equation. Solving for x involves understanding the properties of square roots and the importance of considering both positive and negative solutions. This final step is the culmination of our algebraic journey, where we've successfully unraveled the equation and found the hidden values of x. By mastering this process, you've gained a valuable skill in algebra that will serve you well in more advanced mathematical endeavors.
Conclusion
In conclusion, solving the equation (x+3)(x-3)=16-2x^2 has been a comprehensive journey through the realms of algebra. We began by understanding the problem, dissecting the equation into its components, and strategizing our approach. Each step, from expanding the product to rearranging terms and isolating the variable, was meticulously explained, shedding light on the underlying mathematical principles. We applied the difference of squares identity, rearranged the equation to isolate x^2, and finally, solved for x by taking the square root of both sides. The solutions we found, x = (5√3) / 3 and x = -(5√3) / 3, are the values that satisfy the equation. This exercise highlights the importance of algebraic manipulation, pattern recognition, and careful attention to detail. Solving equations is not just about finding the right answer; it's about developing a deep understanding of mathematical concepts and techniques. By mastering such problems, you enhance your problem-solving skills and gain confidence in your ability to tackle more complex mathematical challenges. Remember, every equation is a puzzle waiting to be solved, and with the right tools and approach, the solution is always within reach. This journey through algebra has not only provided us with the solutions but has also equipped us with the knowledge and skills to navigate the world of equations with greater proficiency and understanding. So, let's continue to explore the fascinating world of mathematics, armed with the confidence and skills we've gained along the way.
