Question

solve:
\frac{x - 3}{x} - \frac{3}{x + 1} + \frac{3}{x^2 + x} = 0
x = ?

Explanation:

Step1: Factor the denominator

Notice that \(x^2 + x = x(x + 1)\), so the common denominator for the fractions is \(x(x + 1)\).
Multiply each term by \(x(x + 1)\) to eliminate the denominators:
\((x - 3)(x + 1) - 3x + 3 = 0\)

Step2: Expand and simplify

Expand \((x - 3)(x + 1)\):
\(x^2 + x - 3x - 3 - 3x + 3 = 0\)
Combine like terms:
\(x^2 - 5x = 0\)

Step3: Factor the quadratic equation

Factor out \(x\):
\(x(x - 5) = 0\)

Step4: Solve for \(x\)

Set each factor equal to zero:
\(x = 0\) or \(x - 5 = 0\)
But \(x = 0\) makes the original denominators zero (undefined), so we discard it.
Solving \(x - 5 = 0\) gives \(x = 5\).

Answer:

\(x = 5\)