Questions: 6/(3x^2-6x) + 4/(3x)

Transcript text: 4) $\frac{6}{3 x^{2}-6 x}+\frac{4}{3 x}$

Solution

Solution Steps

Step 1: Finding the Common Denominator

To simplify the expression $\frac{6}{3x^2 - 6x} + \frac{4}{3x}$, we first factor the denominator of the first fraction. The expression $3x^2 - 6x$ can be factored as $3x(x - 2)$. Thus, the common denominator for both fractions is $3x(x - 2)$.

Step 2: Rewriting the Fractions

Next, we rewrite each fraction with the common denominator: \[ \frac{6}{3x^2 - 6x} = \frac{6}{3x(x - 2)} \quad \text{and} \quad \frac{4}{3x} = \frac{4(x - 2)}{3x(x - 2)} \] Now, we can combine the two fractions: \[ \frac{6 + 4(x - 2)}{3x(x - 2)} \]

Step 3: Simplifying the Expression

We simplify the numerator: \[ 6 + 4(x - 2) = 6 + 4x - 8 = 4x - 2 \] Thus, the combined expression becomes: \[ \frac{4x - 2}{3x(x - 2)} \] This can be factored further: \[ \frac{2(2x - 1)}{3x(x - 2)} \]

Final Answer

The simplified form of the expression $\frac{6}{3x^2 - 6x} + \frac{4}{3x}$ is: \[ \boxed{\frac{2(2x - 1)}{3x(x - 2)}} \]

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