Solve 10/x - 12/(x-3) + 4 = 0

To solve the equation 10/x - 12/(x-3) + 4 = 0, we can follow these steps:

Step 1: Find a common denominator for the fractions on the left side of the equation. In this case, the common denominator for the two fractions is x(x-3).

Step 2: Multiply both sides of the equation by the common denominator x(x-3) to eliminate the fractions. This step helps us simplify the equation and solve for x.

After multiplying, the equation becomes:
10(x-3) - 12x + 4x(x-3) = 0

Step 3: Distribute and simplify the equation.
10x - 30 - 12x + 4x^2 - 12x + 36 = 0
Combine like terms:
4x^2 - 14x + 6 = 0

Step 4: Rearrange the equation into standard quadratic form (ax^2 + bx + c = 0).
4x^2 - 14x + 6 = 0

Step 5: Solve the quadratic equation. You can use the quadratic formula or factoring to find the value(s) of x. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 4, b = -14, and c = 6.

x = (-(-14) ± √((-14)^2 - 4(4)(6))) / (2(4))
x = (14 ± √(196 - 96)) / 8
x = (14 ± √100) / 8
x = (14 ± 10) / 8

Step 6: Simplify the solutions.
x1 = (14 + 10) / 8 = 24 / 8 = 3
x2 = (14 - 10) / 8 = 4 / 8 = 1/2

Therefore, the solutions to the equation 10/x - 12/(x-3) + 4 = 0 are x = 3 and x = 1/2.