Solving the Equation 6/x^2 - 1 + 5 = 8x - 1/4x + 4 - 12x - 1/4 - 4x
In this article, we will solve the given equation step by step.
The Equation
The equation is:
$\frac{6}{x^2} - 1 + 5 = 8x - \frac{1}{4x} + 4 - 12x - \frac{1}{4} - 4x$
Simplifying the Equation
To simplify the equation, we will start by combining like terms:
$\frac{6}{x^2} - 1 + 5 = 8x - \frac{1}{4x} + 4 - 12x - \frac{1}{4} - 4x$
Combine the constant terms:
$\frac{6}{x^2} - 1 + 5 = 8x - 12x - 4x - \frac{1}{4x} - \frac{1}{4} + 4$
Simplify the left-hand side:
$\frac{6}{x^2} + 4 = 8x - 12x - 4x - \frac{1}{4x} - \frac{1}{4} + 4$
Combine the x terms:
$\frac{6}{x^2} + 4 = -8x - \frac{1}{4x} - \frac{1}{4} + 4$
Solving for x
Now, we will try to solve for x.
Multiply both sides by x^2 to eliminate the fraction:
$6 + 4x^2 = -8x^3 - x - \frac{x^2}{4} + 4x^2$
Simplify the equation:
$4x^2 - 8x^3 - x + 6 = 0$
This is a cubic equation in x. Unfortunately, there is no simple formula to solve cubic equations. We need to use numerical methods or algebraic methods such as Cardano's method to solve this equation.
Conclusion
The equation 6/x^2 - 1 + 5 = 8x - 1/4x + 4 - 12x - 1/4 - 4x is a complex equation that cannot be solved easily. We simplified the equation and arrived at a cubic equation, which requires numerical or algebraic methods to solve.
