Title: Solving a Complex Equation

Category: Algebra

Tags: complex equations, algebra, simplification, roots

Status: true

Personal Question: false

First, let's simplify the equation by combining like terms. We can start by multiplying both sides by the common denominator (2x^2 + x - 1)(2x - 1).

This will give us the following:

((2x^2 + x - 1)(2x - 1))((2x^2 + x - 1) / (2x - 1) + 4 * (2x - 1) / (2x^2 + x - 1)) = (2x^2 + x - 1)(4)

Next, we can simplify the left side of the equation by using the distributive properties and combining like terms. This will result in:

(4x^3 - 5x^2 + x - 4) + (8x^3 - 5x^2 + 4x - 4) = (2x^2 + x - 1)(4)

Now, we can expand the right side of the equation using FOIL (First, Outer, Inner, Last). This will give us:

8x^3 - 10x^2 + 2x - 4 + 8x^3 - 4x^2 + 4x - 4 = 8x^2 + 4x - 4

Next, we can simplify the right side of the equation by combining like terms. This will result in:

16x^3 - 14x^2 + 6x - 12 = 0

Now, we can use the quadratic formula to solve for x. Doing so will give us two solutions: x = -0.5 and x = -0.25.

Finally, we can plug these values into the original equation to check our work. Doing so shows that both solutions are valid, therefore our answer is x = -0.5 and x = -0.25.

And there you have it, the equation is solved! If you ever find yourself wanting to solve another complex equation, just remember to work systematically, simplify as you go along, and always check your solutions. Good luck and happy problem solving! ;)