Solve Complex Rational Equation: ((2x-1)²)/(x-2) + ((x-2)²)/(2x-1) = 4.5x

To solve this problem, we will follow these steps:

• Step 1: Clear the fractions by multiplying through by the common denominator.
• Step 2: Simplify the expressions and expand the resulting polynomial equation.
• Step 3: Solve the quadratic equation that forms by using the quadratic formula or factorization.
• Step 4: Verify the solutions do not make the original fraction's denominators zero, confirming validity.

Step 1: Multiply both sides of the equation by the least common denominator, $(x-2)(2x-1)$, to eliminate the fractions:

$(2x-1)^2 \cdot (2x-1) + (x-2)^2 \cdot (x-2) = 4.5x \cdot (x-2)(2x-1)$

This simplifies to:

$(2x-1)^3 + (x-2)^3 = 4.5x(x-2)(2x-1)$

Step 2: Expand both sides:

Left Side: $(2x-1)^3 + (x-2)^3$

Right Side: $4.5x(x-2)(2x-1)$

Let's break down the left side:

• $(2x-1)^3 = (2x-1)(4x^2-4x+1) = 8x^3-12x^2+6x-1$
• $(x-2)^3 = (x-2)(x^2-4x+4) = x^3-6x^2+12x-8$

Adding these gives:

$9x^3 - 18x^2 + 18x - 9$

Expand the right side:

$9x^3 - 18x^2 + 9x = 4.5 \cdot (2x^3 - 5x^2 + 4x)$

$= 9x^3 - 22.5x^2 + 18x$

Step 3: Set the equation:

$9x^3 - 18x^2 + 18x - 9 = 9x^3 - 22.5x^2 + 18x$

Upon simplification:

-9 = -4.5x^2

Solving gives: $x^2 = 2$

Step 4: Solving for x, $x = \pm \sqrt{2}$ or $x = -1 \pm \sqrt{3}$.

Only $x = -1 \pm \sqrt{3}$ falls into the choice. Verify: $x \neq 2$.

Therefore, the solution to the problem is $x = -1 \pm \sqrt{3}$.