Answer :
Sure, let's simplify the given expression step-by-step:
[tex]\[ \frac{3x^2 - 3x}{3x^2 - 3} \][/tex]
1. Factor out the common factors in the numerator and the denominator.
In the numerator \(3x^2 - 3x\), we can factor out a \(3x\):
[tex]\[ 3x^2 - 3x = 3x(x - 1) \][/tex]
In the denominator \(3x^2 - 3\), we can factor out a \(3\):
[tex]\[ 3x^2 - 3 = 3(x^2 - 1) \][/tex]
2. Notice that \(x^2 - 1\) is a difference of squares:
[tex]\[ x^2 - 1 = (x + 1)(x - 1) \][/tex]
So the denominator can also be factored further:
[tex]\[ 3(x^2 - 1) = 3(x + 1)(x - 1) \][/tex]
3. Substitute back the factored forms into the original expression:
[tex]\[ \frac{3x(x - 1)}{3(x + 1)(x - 1)} \][/tex]
4. Cancel out the common factors in the numerator and denominator.
The common factor \(3\) and \((x - 1)\) can be canceled:
[tex]\[ \frac{3x \cancel{(x - 1)}}{3(x + 1)\cancel{(x - 1)}} = \frac{x}{x + 1} \][/tex]
So, the simplified form of the expression \(\frac{3x^2 - 3x}{3x^2 - 3}\) is:
[tex]\[ \frac{x}{x + 1} \][/tex]
Therefore, the correct answer is:
[tex]\[ \text{B. } \frac{x}{x+1} \][/tex]
[tex]\[ \frac{3x^2 - 3x}{3x^2 - 3} \][/tex]
1. Factor out the common factors in the numerator and the denominator.
In the numerator \(3x^2 - 3x\), we can factor out a \(3x\):
[tex]\[ 3x^2 - 3x = 3x(x - 1) \][/tex]
In the denominator \(3x^2 - 3\), we can factor out a \(3\):
[tex]\[ 3x^2 - 3 = 3(x^2 - 1) \][/tex]
2. Notice that \(x^2 - 1\) is a difference of squares:
[tex]\[ x^2 - 1 = (x + 1)(x - 1) \][/tex]
So the denominator can also be factored further:
[tex]\[ 3(x^2 - 1) = 3(x + 1)(x - 1) \][/tex]
3. Substitute back the factored forms into the original expression:
[tex]\[ \frac{3x(x - 1)}{3(x + 1)(x - 1)} \][/tex]
4. Cancel out the common factors in the numerator and denominator.
The common factor \(3\) and \((x - 1)\) can be canceled:
[tex]\[ \frac{3x \cancel{(x - 1)}}{3(x + 1)\cancel{(x - 1)}} = \frac{x}{x + 1} \][/tex]
So, the simplified form of the expression \(\frac{3x^2 - 3x}{3x^2 - 3}\) is:
[tex]\[ \frac{x}{x + 1} \][/tex]
Therefore, the correct answer is:
[tex]\[ \text{B. } \frac{x}{x+1} \][/tex]
