Solve the following equation:
x−2(2x−1)2+2x−1(x−2)2=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⋅(2x−1)+(x−2)2⋅(x−2)=4.5x⋅(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)(4x2−4x+1)=8x3−12x2+6x−1
- (x−2)3=(x−2)(x2−4x+4)=x3−6x2+12x−8
Adding these gives:
9x3−18x2+18x−9
Expand the right side:
9x3−18x2+9x=4.5⋅(2x3−5x2+4x)
=9x3−22.5x2+18x
Step 3: Set the equation:
9x3−18x2+18x−9=9x3−22.5x2+18x
Upon simplification:
-9 = -4.5x^2
Solving gives: x2=2
Step 4: Solving for x, x=±2 or x=−1±3.
Only x=−1±3 falls into the choice. Verify: x=2.
Therefore, the solution to the problem is x=−1±3.