-(3%2B2%2F1-x).jpeg "(2x/2x^2-5x+3-5/2x-3) (3+2/1-x)")
Simplifying the Expression: ((2x)/(2x^2-5x+3) - 5/(2x-3)) (3 + 2/(1-x))
This expression involves multiple fractions and requires careful simplification. Let's break it down step-by-step:
Step 1: Factor the Quadratic Expression
The quadratic expression in the denominator of the first fraction can be factored:
2x² - 5x + 3 = (2x - 3)(x - 1)
Step 2: Find a Common Denominator for the First Fraction
The first fraction has two terms: (2x)/(2x² - 5x + 3) and 5/(2x - 3). To combine them, we need a common denominator:
- (2x)/(2x² - 5x + 3) = (2x)/(2x - 3)(x - 1)
- 5/(2x - 3) = (5(x - 1))/(2x - 3)(x - 1)
Step 3: Combine the First Fraction
Now we can combine the two terms:
(2x)/(2x - 3)(x - 1) - (5(x - 1))/(2x - 3)(x - 1) = (2x - 5(x - 1))/(2x - 3)(x - 1) = (-3x + 5)/(2x - 3)(x - 1)
Step 4: Simplify the Second Fraction
The second fraction is: 3 + 2/(1-x)
To combine them, we need a common denominator:
- 3 = 3(1-x)/(1-x)
- 2/(1-x) = 2/(1-x)
Combining them:
3(1-x)/(1-x) + 2/(1-x) = (3 - 3x + 2)/(1-x) = (5 - 3x)/(1-x)
Step 5: Multiply the Simplified Fractions
Now we have two simplified fractions:
- (-3x + 5)/(2x - 3)(x - 1)
- (5 - 3x)/(1-x)
Multiplying them:
((-3x + 5)/(2x - 3)(x - 1)) * ((5 - 3x)/(1-x)) = (-3x + 5)(5 - 3x) / (2x - 3)(x - 1)(1 - x)
Conclusion
The simplified expression is (-3x + 5)(5 - 3x) / (2x - 3)(x - 1)(1 - x). This form can be further expanded but is considered simplified in its current form.