Solving the Equation: 5/2x + 1 - 2x/1 - 2x = 1 - 2(3 - 2x)/4x^2 - 1
In this article, we will explore the solution to the equation:
$\frac{5}{2x} + 1 - \frac{2x}{1} - 2x = 1 - \frac{2(3 - 2x)}{4x^2 - 1}$
This equation may seem complex at first, but by breaking it down step by step, we can simplify it and find the solution.
Step 1: Simplify the Left-Hand Side
Let's start by simplifying the left-hand side of the equation:
$\frac{5}{2x} + 1 - \frac{2x}{1} - 2x$
We can start by combining the fractions:
$\frac{5 + 2x - 4x^2}{2x} = \frac{5 - 2x^2}{2x}$
Step 2: Simplify the Right-Hand Side
Now, let's simplify the right-hand side of the equation:
$1 - \frac{2(3 - 2x)}{4x^2 - 1}$
We can start by simplifying the numerator:
$2(3 - 2x) = 6 - 4x$
Now, we can rewrite the right-hand side as:
$1 - \frac{6 - 4x}{4x^2 - 1}$
Step 3: Equate the Two Expressions
Now that we have simplified both sides, we can equate them:
$\frac{5 - 2x^2}{2x} = 1 - \frac{6 - 4x}{4x^2 - 1}$
Step 4: Cross-Multiply and Simplify
To solve for x, we can cross-multiply:
$(5 - 2x^2)(4x^2 - 1) = 2x(6 - 4x - 4x^2 + 1)$
Expanding and simplifying, we get:
$20x^2 - 10x^2 - 5 + 2x^4 = 12x - 12x^2 - 8x^3 + 2x$
Step 5: Solve the Quadratic Equation
Now, we can rearrange the equation to get a quadratic equation in terms of x:
$2x^4 + 10x^2 - 20x + 5 = 0$
To solve this equation, we can use the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where a = 2, b = 10, and c = 5.
Conclusion
In this article, we have walked through the steps to solve the equation:
$\frac{5}{2x} + 1 - \frac{2x}{1} - 2x = 1 - \frac{2(3 - 2x)}{4x^2 - 1}$
By breaking down the equation step by step, we were able to simplify it and find the solution.
