Solving The Equation 5/(2+x) + 2/4 = 1/(2-x) For X
Hey guys! Today, we're diving into a cool math problem where we need to find the value of x in a given equation. This kind of problem is super common in algebra, and it's a great way to sharpen our skills in handling fractions and algebraic manipulations. So, let's break it down step by step and make sure we understand every little detail. Get ready to put on your thinking caps, because we're about to solve this equation together!
Understanding the Problem
First off, let's take a good look at the equation we're dealing with:
5/(2+x) + 2/4 = 1/(2-x)
This equation involves fractions with x in the denominator, which means we need to be extra careful with our algebraic steps. Our mission is to isolate x on one side of the equation. To do that, we'll need to clear the fractions, simplify the equation, and then solve for x. It might sound like a lot, but don't worry! We'll tackle it one step at a time. Trust me, by the end of this, you'll feel like a math whiz!
Before we jump into the calculations, it’s important to note that we need to consider any values of x that would make the denominators zero, as those values would make the fractions undefined. So, we need to keep in mind that x cannot be -2 or 2. This is a crucial point to remember as we move forward. Understanding these restrictions will help us avoid potential pitfalls and ensure we arrive at the correct solution. Okay, let's get started!
Initial Equation
Our initial equation is:
5/(2+x) + 2/4 = 1/(2-x)
We need to find the value of x that satisfies this equation. The first thing we're going to do is simplify the term 2/4. This will make our equation a bit easier to handle. Simplifying fractions is a fundamental skill, and it's always a good idea to do it whenever you can. It helps keep the numbers smaller and the calculations simpler. So, let's simplify 2/4 to its simplest form.
Simplify 2/4
The fraction 2/4 can be simplified to 1/2. So, let's replace 2/4 with 1/2 in our equation. This simple step will make our equation a bit cleaner and easier to work with. Remember, simplifying fractions is all about dividing both the numerator and the denominator by their greatest common divisor. In this case, the greatest common divisor of 2 and 4 is 2. Dividing both by 2 gives us 1/2. It's these little simplifications that can make a big difference in the long run, especially when dealing with more complex equations.
Simplified Equation
Now our equation looks like this:
5/(2+x) + 1/2 = 1/(2-x)
This looks a bit better already, doesn't it? We've taken the first step towards solving for x. Now, we need to clear the fractions to make the equation easier to manipulate. This is a common strategy in algebra when dealing with equations involving fractions. To clear the fractions, we'll need to find the least common denominator (LCD) of the fractions in the equation. Once we have the LCD, we'll multiply every term in the equation by it. This will eliminate the denominators and give us a simpler equation to solve. So, let's move on to finding the LCD.
Clearing the Fractions
To clear the fractions, we need to find the least common denominator (LCD) of the denominators in our equation. Our denominators are (2+x), 2, and (2-x). The LCD will be the smallest expression that each of these denominators can divide into evenly.
Finding the LCD
The denominators are (2+x), 2, and (2-x). The LCD is the product of these distinct factors, which is 2(2+x)(2-x). Finding the LCD is a crucial step because it allows us to eliminate the fractions and work with a simpler equation. The LCD is essentially the smallest multiple that all the denominators share. Once we have the LCD, we can multiply each term in the equation by it, which will cancel out the denominators and leave us with a more manageable equation. This is a standard technique in algebra, and mastering it is key to solving equations with fractions.
Multiply by LCD
We'll multiply each term in the equation by the LCD, which is 2(2+x)(2-x):
2(2+x)(2-x) * [5/(2+x)] + 2(2+x)(2-x) * [1/2] = 2(2+x)(2-x) * [1/(2-x)]
Multiplying each term by the LCD is a fundamental step in clearing the fractions. This step ensures that we maintain the equality of the equation while eliminating the denominators. When we multiply, we distribute the LCD to each term, and then we cancel out the common factors. This process transforms the equation into a form that's much easier to solve. It's like clearing away the clutter so we can see the path to the solution more clearly. So, let's go through the multiplication and cancellation carefully.
Cancelling Terms
Now, let's cancel the common terms in each part:
2(2-x) * 5 + (2+x)(2-x) = 2(2+x)
Notice how the denominators have been cleared? This is the magic of multiplying by the LCD! By cancelling out the common factors, we've transformed our equation from one involving fractions to a much simpler algebraic equation. This is a significant step forward, as we can now focus on expanding, simplifying, and solving for x. Clearing the fractions makes the rest of the process much smoother and less prone to errors. So, let's move on to the next step and continue simplifying the equation.
Simplifying the Equation
Now we have a simpler equation:
10(2-x) + (2+x)(2-x) = 2(2+x)
Expanding the Terms
Let's expand the terms:
20 - 10x + (4 - x^2) = 4 + 2x
Expanding the terms involves multiplying out the expressions to remove the parentheses. This is a crucial step in simplifying the equation. We need to be careful with the signs and make sure we multiply each term correctly. Expanding the terms allows us to combine like terms and move towards isolating x. It's like untangling a knot, where each step brings us closer to a clearer and simpler form. So, let's double-check our expansions and make sure we haven't missed anything.
Combine Like Terms
Now, let's combine the like terms and rearrange the equation:
20 - 10x + 4 - x^2 = 4 + 2x
24 - 10x - x^2 = 4 + 2x
Combining like terms is a fundamental step in simplifying any algebraic equation. It involves grouping together terms that have the same variable and exponent. This makes the equation cleaner and easier to work with. By combining like terms, we reduce the number of terms in the equation, which simplifies the subsequent steps. It's like organizing your workspace before starting a project; a clear and organized equation is much easier to solve. So, let's make sure we've combined all the like terms correctly.
Rearranging the Equation
Move all terms to one side to set the equation to zero:
0 = x^2 + 12x - 20
Setting the equation to zero is a common strategy when dealing with quadratic equations. This step allows us to use techniques like factoring or the quadratic formula to find the solutions for x. By moving all the terms to one side, we create a standard form of the quadratic equation, which is ax² + bx + c = 0. This form is essential for applying the quadratic formula or attempting to factor the equation. It's like preparing the canvas before painting; setting the equation to zero sets the stage for the next steps in solving for x.
Solving the Quadratic Equation
We have a quadratic equation:
x^2 + 12x - 20 = 0
Using the Quadratic Formula
Since this quadratic equation doesn't factor easily, we'll use the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
Where a = 1, b = 12, and c = -20.
The quadratic formula is a powerful tool for solving quadratic equations, especially when they don't factor easily. It provides a direct method for finding the solutions for x. The formula is derived from the process of completing the square and is a fundamental concept in algebra. By using the quadratic formula, we can solve any quadratic equation, regardless of its complexity. It's like having a universal key that unlocks the solutions to any quadratic equation. So, let's plug in the values and calculate the solutions.
Plugging in the Values
Plugging in the values, we get:
x = [-12 ± sqrt(12^2 - 4 * 1 * (-20))] / (2 * 1)
x = [-12 ± sqrt(144 + 80)] / 2
x = [-12 ± sqrt(224)] / 2
Plugging in the values into the quadratic formula is a critical step. We need to be careful with the signs and the order of operations to ensure we get the correct result. This step involves substituting the coefficients a, b, and c into the formula and then simplifying the expression. It's like following a recipe; each ingredient needs to be added in the correct amount and at the right time. So, let's double-check our substitutions and calculations to make sure we haven't made any errors.
Simplifying the Square Root
Simplifying the square root:
x = [-12 ± sqrt(16 * 14)] / 2
x = [-12 ± 4sqrt(14)] / 2
Simplifying the square root is an important step in getting the solutions in their simplest form. We look for perfect square factors within the square root and take them out. This makes the expression cleaner and easier to work with. Simplifying radicals is a fundamental skill in algebra, and it's often necessary to present the solutions in their most concise form. It's like polishing a gem to reveal its brilliance; simplifying the square root reveals the underlying simplicity of the solution.
Final Solutions
Divide both terms in the numerator by 2:
x = -6 ± 2sqrt(14)
So, the two possible values for x are:
x = -6 + 2sqrt(14)
x = -6 - 2sqrt(14)
These are the exact solutions for x. However, since the options given are integers, we need to approximate these values to see which integer option matches.
Approximating the Solutions
Approximate sqrt(14)
We know that sqrt(14) is between sqrt(9) = 3 and sqrt(16) = 4. A closer approximation is around 3.7.
Approximating the square root is necessary when we need to compare our solutions to integer options or when we need a decimal value for practical purposes. It involves estimating the value of the square root without using a calculator. This can be done by knowing the perfect squares around the number and making an educated guess. Approximating the square root helps us bridge the gap between the exact solution and a more practical or comparable value. It's like estimating the distance to a destination; it gives us a sense of where we're going even if we don't have the exact measurement.
Approximate x Values
So,
x ≈ -6 + 2 * 3.7 = -6 + 7.4 = 1.4
x ≈ -6 - 2 * 3.7 = -6 - 7.4 = -13.4
Comparing with Options
The first approximate value, 1.4, is close to 1, which is one of the options. The second value, -13.4, is not close to any of the options.
Comparing our approximate solutions with the given options is the final step in identifying the correct answer. We look for the option that is closest to our calculated values. This step involves judgment and an understanding of the level of approximation. It's like matching a key to a lock; we're looking for the option that fits the solution we've found. By comparing the values, we can confidently select the correct answer.
Final Answer
Therefore, the closest integer value for x is 1.
So, the correct answer is C. 1.
Conclusion
Woohoo! We did it! We successfully solved the equation by clearing fractions, simplifying, and using the quadratic formula. This problem was a fantastic workout for our algebra muscles, and I hope you feel a sense of accomplishment. Remember, math isn't just about finding the right answer; it's about the journey of problem-solving and the skills you develop along the way. Keep practicing, and you'll become a math master in no time! Keep up the awesome work, guys!
