Question

(2x+3)/(x² -4x+4) - (x-1)/(x²-2x)=5/x

Answer 1

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Answer 2

To solve the given equation, we need to find the value(s) of 'x' that make the equation true. Let's proceed step by step.

Given equation: (2x + 3)/(x² - 4x + 4) - (x - 1)/(x² - 2x) = 5/x

Step 1: Factorize the denominators: x² - 4x + 4 = (x - 2)² x² - 2x = x(x - 2)

Step 2: Find a common denominator for both fractions. The common denominator will be x(x - 2)(x - 2).

Step 3: Rewrite the equation with the common denominator: [(2x + 3)(x - 2)(x - 2)]/[x(x - 2)(x - 2)] - [(x - 1)(x - 2)(x - 2)]/[x(x - 2)(x - 2)] = 5/x

Step 4: Combine the fractions on the left side of the equation: [(2x + 3)(x - 2)(x - 2) - (x - 1)(x - 2)(x - 2)]/[x(x - 2)(x - 2)] = 5/x

Step 5: Expand and simplify the numerators: [2x(x - 2)(x - 2) + 3(x - 2)(x - 2) - (x - 1)(x - 2)(x - 2)]/[x(x - 2)(x - 2)] = 5/x

Step 6: Distribute and simplify further: [2x(x² - 4x + 4) + 3(x² - 4x + 4) - (x³ - 4x² + 4x - 2x² + 8x - 8)]/[x(x - 2)(x - 2)] = 5/x

Step 7: Expand and combine like terms in the numerator: [2x³ - 8x² + 8x + 3x² - 12x + 12 - x³ + 6x² - 8x]/[x(x - 2)(x - 2)] = 5/x

Step 8: Combine like terms in the numerator: [x³ + x² - 12x + 12]/[x(x - 2)(x - 2)] = 5/x

Step 9: Multiply both sides by x(x - 2)(x - 2) to eliminate the denominators: x³ + x² - 12x + 12 = 5(x - 2)(x - 2)

Step 10: Expand the right side: x³ + x² - 12x + 12 = 5(x² - 4x + 4)

Step 11: Distribute on the right side: x³ + x² - 12x + 12 = 5x² - 20x + 20

Step 12: Move all terms to one side to set the equation to zero: x³ + x² - 12x + 12 - 5x² + 20x - 20 = 0

Step 13: Combine like terms: x³ - 4x² + 8x - 8 = 0

At this point, we have a cubic equation in 'x'. Unfortunately, finding exact solutions for cubic equations can be challenging. However, it seems that it might be possible to factor this equation or use numerical methods to approximate the solutions. Alternatively, if this equation is part of a larger problem or has specific constraints, those constraints might lead to a simpler solution.