Question

Simplify the expression. $$\frac{9}{x^{2}-3 x}+\frac{3}{x-3}$$

Short answer

The simplified form of the expression is \( \frac{3x + 9}{x^2 - 3x}\)

Step-by-step solution

Factor the denominator of the first fraction

Rewrite the denominator of the first fraction as \(x(x - 3)\) from \(x^2 - 3x\). Thereby, the first fraction becomes \(\frac{9}{x(x - 3)}\).

Find a common denominator

A common denominator for the fractions \(\frac{9}{x(x - 3)}\) and \(\frac{3}{x - 3}\) would be \(x(x - 3)\), because it incorporates all the distinct factors in the original denominators. Therefore, multiply the second fraction by \(x\) at both top and bottom, which yields \(\frac{3x}{x(x - 3)}\).

Combine the fractions

Now, both fractions have the same denominator, and they can be combined: \(\frac{9}{x(x - 3)} + \frac{3x}{x(x - 3)} = \frac{9 + 3x}{x(x - 3)}\).

Simplify the result

Rearrange terms in the numerator to get a final answer, so \(\frac{9 + 3x}{x(x - 3)}\) becomes \(\frac{3x + 9}{x^2 - 3x}\)