Questions: Simplify the expression (3x+9)/(3x^2+12x+9) (5x^2-5x)/(x^2-1)

Transcript text: Simplify the expression $\frac{3 x+9}{3 x^{2}+12 x+9}$ $\frac{5 x^{2}-5 x}{x^{2}-1}$

Solution

Solution Steps

To simplify the given expression, we need to factorize both the numerator and the denominator of each fraction and then simplify by canceling out common factors.

Step 1: Factorize the Numerator and Denominator

First, we factorize the numerator and the denominator of each fraction: \[ \frac{3x + 9}{3x^2 + 12x + 9} \cdot \frac{5x^2 - 5x}{x^2 - 1} \]

Step 2: Simplify Each Fraction

Factorize the expressions: \[ 3x + 9 = 3(x + 3) \] \[ 3x^2 + 12x + 9 = 3(x^2 + 4x + 3) = 3(x + 3)(x + 1) \] \[ 5x^2 - 5x = 5x(x - 1) \] \[ x^2 - 1 = (x - 1)(x + 1) \]

Step 3: Substitute and Simplify

Substitute the factored forms back into the expression: \[ \frac{3(x + 3)}{3(x + 3)(x + 1)} \cdot \frac{5x(x - 1)}{(x - 1)(x + 1)} \]

Cancel out the common factors: \[ \frac{1}{(x + 1)} \cdot \frac{5x}{(x + 1)} = \frac{5x}{(x + 1)^2} \]