Solve X - √(16 - X²) = 4: A Detailed Guide
Hey guys! Today, we're diving deep into the fascinating world of algebra to tackle a seemingly complex equation. Don't worry, we'll break it down step by step so that everyone can follow along. Our mission? To solve the equation x - √(16 - x²) = 4. Sounds intimidating? Trust me, with a little algebraic maneuvering, we can conquer it. So, grab your pencils, notebooks, and let's get started!
1. Isolating the Radical
The first rule of fight club… I mean, the first step in solving equations with square roots is to isolate the radical. This means getting the square root term all by itself on one side of the equation. In our case, we have x - √(16 - x²) = 4. To isolate the square root, we need to move the x term to the right side. We can do this by subtracting x from both sides of the equation. This gives us: -√(16 - x²) = 4 - x. Now, to make things a bit easier (and get rid of that pesky negative sign on the square root), let's multiply both sides of the equation by -1. This will flip the signs on both sides, resulting in: √(16 - x²) = x - 4. Ta-da! We've successfully isolated the radical. This is a crucial step because it allows us to get rid of the square root in the next step. Remember, isolating the radical is like setting the stage for the main event – squaring both sides.
Isolating the radical is a critical initial step because it sets the stage for eliminating the square root. By having the radical term alone on one side of the equation, we can confidently square both sides without introducing unnecessary complexity. In this specific equation, x - √(16 - x²) = 4, the presence of the term x alongside the square root initially complicates the process of solving for x. Moving x to the right side allows us to deal with the square root directly. The manipulation of multiplying both sides by -1 is a clever trick to simplify the equation further. This step ensures that we are working with a positive square root, which makes the subsequent squaring operation more straightforward and less prone to sign errors. The goal here is not just to isolate the radical, but to also prepare the equation for the next phase of the solution. The isolated radical, √(16 - x²) = x - 4, is now in a form that allows us to apply the squaring operation effectively, leading us closer to finding the values of x that satisfy the original equation. This methodical approach, breaking down the problem into manageable steps, is key to solving more complex algebraic equations. So, with the radical isolated, we are well-prepared to move on to the next stage: eliminating the square root by squaring both sides.
2. Squaring Both Sides
Alright, we've got the radical isolated! Now comes the fun part – squaring both sides. Why do we do this? Because squaring a square root cancels it out, leaving us with just the expression inside the square root. So, we have √(16 - x²) = x - 4. When we square both sides, we get (√(16 - x²))² = (x - 4)². The left side simplifies beautifully: the square root and the square cancel each other out, leaving us with 16 - x². The right side, however, needs a little more attention. Remember that (x - 4)² is not the same as x² - 4². We need to expand it using the good old FOIL method (First, Outer, Inner, Last) or the binomial square formula: (a - b)² = a² - 2ab + b². Applying this to our equation, we get (x - 4)² = x² - 8x + 16. Now, our equation looks like this: 16 - x² = x² - 8x + 16. We've successfully eliminated the square root, and we're left with a quadratic equation. This is progress, guys! We're one step closer to finding the solution.
Squaring both sides of the equation is a fundamental technique for eliminating square roots, but it's crucial to understand the implications of this operation. When we square both sides, we are essentially saying that if two quantities are equal, their squares are also equal. However, it's important to remember that this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. This happens because squaring can make two unequal quantities equal (e.g., -2 and 2 both square to 4). Therefore, after squaring, it's essential to check our solutions in the original equation to ensure they are valid. In our case, squaring √(16 - x²) = x - 4 resulted in 16 - x² = (x - 4)². The simplification of the left side is straightforward, but the expansion of the right side, (x - 4)², requires careful attention. As we noted, it's a common mistake to assume (x - 4)² = x² - 4². The correct expansion, using the binomial square formula, yields x² - 8x + 16. This highlights the importance of mastering algebraic identities and being meticulous with algebraic manipulations. The resulting equation, 16 - x² = x² - 8x + 16, is a quadratic equation, which is a significant step forward because we have well-established methods for solving quadratic equations. However, the journey isn't over yet. We must proceed to solve the quadratic equation and then, crucially, verify the solutions in the original equation to rule out any extraneous solutions. So, while squaring both sides is a powerful tool, it comes with a responsibility to check our work and ensure the validity of our solutions.
3. Simplifying to a Quadratic Equation
Now that we've squared both sides, we have the equation 16 - x² = x² - 8x + 16. Our next goal is to simplify this into a standard quadratic equation form, which looks like ax² + bx + c = 0. To do this, we need to get all the terms on one side of the equation. Let's add x² to both sides: 16 = 2x² - 8x + 16. Next, let's subtract 16 from both sides: 0 = 2x² - 8x. Awesome! We've successfully transformed our equation into a quadratic form. Notice that in this case, our c term is 0, which will make our lives a little easier when we go to solve it.
Simplifying the equation to a standard quadratic form is a crucial step in solving for x. The standard form, ax² + bx + c = 0, provides a clear structure that allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. The process of rearranging the terms in our equation, 16 - x² = x² - 8x + 16, involves strategically adding and subtracting terms to consolidate them on one side. This is not just a mechanical process; it's about creating a recognizable and manageable form. Adding x² to both sides and then subtracting 16 from both sides are deliberate steps aimed at achieving the standard quadratic form. The resulting equation, 0 = 2x² - 8x, is a simplified quadratic equation where the constant term c is zero. This particular form is advantageous because it simplifies the factoring process, as we will see in the next step. The ability to manipulate equations and transform them into more convenient forms is a hallmark of algebraic proficiency. By bringing the equation into the standard quadratic form, we have opened the door to a range of solution techniques, making the problem significantly more tractable. The next step involves applying one of these techniques, in this case, factoring, to find the values of x that satisfy the equation. So, simplifying to the standard quadratic form is not just about aesthetics; it's about enabling us to apply powerful problem-solving tools.
4. Solving the Quadratic Equation
We've got our quadratic equation: 0 = 2x² - 8x. Now, how do we solve it? The easiest way to solve this particular quadratic is by factoring. Notice that both terms have a common factor of 2x. Let's factor that out: 0 = 2x(x - 4). Now we have a product of two factors that equals zero. This means that at least one of the factors must be zero. So, we have two possibilities:
- 2x = 0
- x - 4 = 0
Solving the first equation, 2x = 0, we divide both sides by 2 to get x = 0. Solving the second equation, x - 4 = 0, we add 4 to both sides to get x = 4. So, we have two potential solutions: x = 0 and x = 4. But remember, we squared both sides earlier, which means we might have introduced extraneous solutions. We need to check these solutions in the original equation.
Solving the quadratic equation is a critical step in finding the values of x that satisfy the equation, but the method we choose can significantly impact the efficiency and accuracy of the solution. In this case, the quadratic equation 0 = 2x² - 8x is particularly well-suited for factoring due to the absence of a constant term and the presence of a common factor in both terms. Factoring is often the quickest and most straightforward method when applicable, as it avoids the more complex calculations involved in the quadratic formula or completing the square. The key to factoring is recognizing the greatest common factor, which in this case is 2x. By factoring out 2x, we transform the equation into 0 = 2x(x - 4). This form is incredibly powerful because it leverages the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This allows us to split the quadratic equation into two simpler linear equations: 2x = 0 and x - 4 = 0. Solving these linear equations is straightforward and yields the potential solutions x = 0 and x = 4. However, it's crucial to emphasize that these are only potential solutions at this stage. As we discussed earlier, squaring both sides of an equation can introduce extraneous solutions, so we must verify these values in the original equation to ensure they are valid. The factoring approach not only simplifies the solution process but also provides a clear and intuitive way to understand the roots of the quadratic equation. So, while we have found potential solutions, the next step, checking for extraneous solutions, is equally important to ensure the integrity of our answer.
5. Checking for Extraneous Solutions
This is a super important step, guys! We can't just assume that the solutions we found are correct. We need to plug them back into the original equation to make sure they work. Our original equation was x - √(16 - x²) = 4. Let's check our solutions one by one.
Checking x = 0:
Substitute x = 0 into the original equation: 0 - √(16 - 0²) = 4. This simplifies to -√16 = 4, which further simplifies to -4 = 4. This is definitely not true. So, x = 0 is an extraneous solution. It's a pretender! It looks like a solution, but it doesn't actually satisfy the original equation.
Checking x = 4:
Substitute x = 4 into the original equation: 4 - √(16 - 4²) = 4. This simplifies to 4 - √(16 - 16) = 4, which further simplifies to 4 - √0 = 4. This becomes 4 - 0 = 4, which is 4 = 4. This is absolutely true! So, x = 4 is a valid solution.
We've done it! We found one valid solution and identified an extraneous one. Checking for extraneous solutions is a critical step in solving equations with radicals because the process of squaring both sides can introduce solutions that don't satisfy the original equation. This is due to the fact that squaring both a positive and a negative number results in a positive number. Therefore, when we square both sides, we are potentially including solutions that would only work if one side of the original equation was negative. The original equation, x - √(16 - x²) = 4, has a square root term, which by definition represents the non-negative square root. This constraint is crucial in determining the validity of our solutions. When we checked x = 0, we found that it led to the equation -√16 = 4, which simplifies to -4 = 4. This is a clear contradiction, indicating that x = 0 is an extraneous solution. The negative sign in front of the square root is a key factor here. On the other hand, when we checked x = 4, we found that it satisfied the original equation, confirming its validity. The process of checking for extraneous solutions highlights the importance of understanding the properties of the mathematical operations we perform and the potential consequences they can have on the solution set. It's not enough to just find potential solutions; we must rigorously verify them to ensure they are genuine solutions to the original problem. So, by carefully checking our solutions, we can confidently state that x = 4 is the only valid solution to the equation.
6. The Final Solution
After all that work, we've arrived at our final answer! The only valid solution to the equation x - √(16 - x²) = 4 is x = 4. We successfully navigated through isolating the radical, squaring both sides, simplifying to a quadratic, solving the quadratic, and, most importantly, checking for extraneous solutions. This problem showcases the importance of following each step carefully and understanding why we do what we do. Algebra isn't just about memorizing formulas; it's about understanding the underlying logic and principles. And you guys totally crushed it! Great job!
We've successfully solved the equation x - √(16 - x²) = 4, and the journey has underscored several important principles in algebra. The final solution, x = 4, is not just a number; it's the culmination of a series of carefully executed steps and a deep understanding of algebraic techniques. The process began with isolating the radical, a critical first step that set the stage for eliminating the square root. Squaring both sides followed, a powerful tool but one that demanded caution due to the potential for introducing extraneous solutions. The simplification to a quadratic equation demonstrated the importance of algebraic manipulation and the ability to transform equations into more manageable forms. Solving the quadratic equation by factoring highlighted the efficiency of this method when applicable and the significance of the zero-product property. However, the most crucial step in this process was checking for extraneous solutions. This step is not merely a formality; it's a fundamental aspect of solving equations with radicals and a testament to the importance of rigor in mathematics. By plugging our potential solutions back into the original equation, we ensured their validity and discarded the extraneous solution x = 0. This process reinforces the idea that solving equations is not just about finding numbers that satisfy a transformed equation; it's about finding numbers that satisfy the original equation and the constraints it imposes. The solution x = 4 represents the only value that truly makes the original equation true. This problem serves as a valuable example of how a systematic approach, combined with a thorough understanding of algebraic principles, can lead to a successful solution, even for seemingly complex equations. So, by mastering these techniques and concepts, we can tackle a wide range of algebraic challenges with confidence.
Solve the equation x - √(16 - x²) = 4: a detailed, step-by-step solution.
Solve x - √(16 - x²) = 4: Step-by-Step Solution
