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

Question

Solve the following equation:

(2x1)2x2+(x2)22x1=4.5x \frac{(2x-1)^2}{x-2}+\frac{(x-2)^2}{2x-1}=4.5x

Video Solution

Solution Steps

00:00 Solve
00:07 Multiply by denominators to eliminate fractions
00:17 Convert cube to square and product
00:29 Expand brackets using short multiplication formulas
00:55 Properly expand brackets, each term multiplies each term
01:19 Collect like terms
01:46 Arrange the equation so one side equals 0
01:52 Divide by half so the coefficient of X squared is 1
01:55 Use the roots formula to find the appropriate solutions

Step-by-Step Solution

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, (x2)(2x1)(x-2)(2x-1), to eliminate the fractions:

(2x1)2(2x1)+(x2)2(x2)=4.5x(x2)(2x1) (2x-1)^2 \cdot (2x-1) + (x-2)^2 \cdot (x-2) = 4.5x \cdot (x-2)(2x-1)

This simplifies to:

(2x1)3+(x2)3=4.5x(x2)(2x1) (2x-1)^3 + (x-2)^3 = 4.5x(x-2)(2x-1)

Step 2: Expand both sides:

Left Side: (2x1)3+(x2)3(2x-1)^3 + (x-2)^3

Right Side: 4.5x(x2)(2x1)4.5x(x-2)(2x-1)

Let's break down the left side:

  • (2x1)3=(2x1)(4x24x+1)=8x312x2+6x1(2x-1)^3 = (2x-1)(4x^2-4x+1) = 8x^3-12x^2+6x-1
  • (x2)3=(x2)(x24x+4)=x36x2+12x8(x-2)^3 = (x-2)(x^2-4x+4) = x^3-6x^2+12x-8

Adding these gives:

9x318x2+18x99x^3 - 18x^2 + 18x - 9

Expand the right side:

9x318x2+9x=4.5(2x35x2+4x)9x^3 - 18x^2 + 9x = 4.5 \cdot (2x^3 - 5x^2 + 4x)

=9x322.5x2+18x= 9x^3 - 22.5x^2 + 18x

Step 3: Set the equation:

9x318x2+18x9=9x322.5x2+18x9x^3 - 18x^2 + 18x - 9 = 9x^3 - 22.5x^2 + 18x

Upon simplification:

-9 = -4.5x^2

Solving gives: x2=2x^2 = 2

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

Only x=1±3 x = -1 \pm \sqrt{3} falls into the choice. Verify: x2 x \neq 2.

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

Answer

1±3 -1\pm\sqrt{3}