Which of the following is equivalent to `(x ^(2) -10x + 25)/(3x ^(2)-75)` ? `

Options

A. `(3 (x-5))/((x+5))`
B. `(3(x +5))/((x -5))`
C. `((x-5))/(3 (x +5))`
D. `((x +5))/(3 (x-5))`

Explanation

To solve the expression \((x^2 - 10x + 25)/(3x^2 - 75)\), we will simplify it step by step.

### Step 1: Factor the Numerator
The numerator \(x^2 - 10x + 25\) can be recognized as a perfect square trinomial. It can be factored as:
\[ x^2 - 10x + 25 = (x - 5)^2 \]

### Step 2: Factor the Denominator
The denominator \(3x^2 - 75\) can be factored by taking out the common factor of 3:
\[ 3x^2 - 75 = 3(x^2 - 25) \]
Next, we can recognize that \(x^2 - 25\) is a difference of squares, which can be factored as:
\[ x^2 - 25 = (x - 5)(x + 5) \]
Thus, the denominator becomes:
\[ 3(x^2 - 25) = 3(x - 5)(x + 5) \]

### Step 3: Rewrite the Expression
Now we can rewrite the original expression using the factored forms:
\[ \frac{(x - 5)^2}{3(x - 5)(x + 5)} \]

### Step 4: Cancel Common Factors
We can cancel the common factor \((x - 5)\) from the numerator and denominator:
\[ \frac{(x - 5)}{3(x + 5)} \]

### Final Result
Thus, the simplified expression is:
\[ \frac{x - 5}{3(x + 5)} \]