%5E(2)(x%5E(2)-7x%2B10)(2x-1)%5E(2))%2F((x-1)%5E(5)(2x-3)%5E(2))-%3D0.jpeg "(x(x-1)^(2)(x^(2)-7x+10)(2x-1)^(2))/((x-1)^(5)(2x-3)^(2)) =0")
Solving the Equation: (x(x-1)^(2)(x^(2)-7x+10)(2x-1)^(2))/((x-1)^(5)(2x-3)^(2)) = 0
This equation presents a fascinating challenge in algebra. We'll break down the steps to solve it and understand the solutions.
1. Simplifying the Expression
First, we can simplify the given expression by canceling out common factors in the numerator and denominator:
-
Factor the quadratic in the numerator: (x^2 - 7x + 10) = (x-5)(x-2)
-
Cancel common factors: (x-1)^2 appears in both numerator and denominator. We can cancel it, leaving (x-1)^3 in the denominator.
The simplified expression becomes: (x(x-2)(x-5)(2x-1)^2) / ((x-1)^3 (2x-3)^2)
2. Finding the Roots
For the entire expression to equal zero, the numerator must equal zero. Therefore, we need to find the values of x that make the numerator zero.
- Set each factor in the numerator to zero:
- x = 0
- x - 2 = 0 => x = 2
- x - 5 = 0 => x = 5
- 2x - 1 = 0 => x = 1/2
3. Excluding Values that Make the Denominator Zero
The denominator can't be zero, as it would result in division by zero. We need to exclude any values of x that make the denominator zero:
- Set each factor in the denominator to zero:
- x - 1 = 0 => x = 1
- 2x - 3 = 0 => x = 3/2
4. Solutions
The solutions to the equation are the values of x that make the numerator zero and do not make the denominator zero:
Therefore, the solutions to the equation are: x = 0, x = 2, x = 5, and x = 1/2.