Perform the following operation and express in simplest form.
start fraction, x, plus, 3, divided by, x, plus, 2, end fraction, dot, start fraction, x, squared, minus, x, minus, 6, divided by, x, squared, minus, 9, end fraction
x+2
x+3
⋅
x
2
−9
x
2
−x−6
To simplify the expression, let's rewrite it step by step:
(x + 3)/(x + 2) * (x^2 - x - 6)/(x^2 - 9)
First, let's factor the numerators and denominators:
(x + 3)/(x + 2) * (x - 3)(x + 2)/(x + 3)(x - 3)
Next, we can cancel out the common factors in the numerator and denominator:
(x + 3)/(x + 3) * (x - 3)/(x - 3)
We are left with:
1 * 1 = 1
So the expression simplifies to 1.
To simplify the expression, let's step-by-step perform the given operation:
1. Step: Simplify the numerator of the first fraction by adding x and 3:
x + 3
2. Step: Simplify the denominator of the first fraction by adding 2 to x:
x + 2
So, the first fraction becomes (x + 3) / (x + 2).
3. Step: Simplify the numerator of the second fraction by factoring:
x^2 - x - 6 = (x - 3)(x + 2)
4. Step: Simplify the denominator of the second fraction by factoring:
x^2 - 9 = (x - 3)(x + 3)
So, the second fraction becomes (x - 3)(x + 2) / (x - 3)(x + 3).
5. Step: Now we can cancel out the common factors in the numerator and the denominator:
(x + 3)(x - 3) gets canceled out.
So, the simplified expression is (x + 2) / (x + 3).
To simplify the expression, we can divide both the numerator and denominator separately.
Let's start by fractionating the expression:
\[
\frac{\left(\frac{x+3}{x+2}\right) \cdot \left(\frac{x^2-x-6}{x^2-9}\right)}{x}
\]
Now let's simplify each part separately:
First, let's simplify the numerator:
\[
\frac{x+3}{x+2} \cdot \frac{x^2-x-6}{x^2-9}
\]
The numerator can be factored as follows:
\[
(x+3)(x-2)
\]
The denominator can also be factored as follows:
\[
(x+3)(x-3)
\]
Now, we can cancel out the common factor (x+3):
\[
\frac{(x+3)(x-2)}{(x+3)(x-3)}
\]
The (x+3) factors in the numerator and denominator cancel out:
\[
\frac{x-2}{x-3}
\]
Next, let's simplify the expression by dividing the result by x:
\[
\frac{x-2}{x-3} \cdot \frac{1}{x}
\]
We can rewrite this expression as:
\[
\frac{x-2}{x(x-3)}
\]
So, the simplified form of the expression is:
\[
\frac{x-2}{x(x-3)}
\]