Simplify : `(10x)/(5+x)`

Answer

To simplify the expression \(\frac{10x}{5+x}\), we will follow these steps:

### Step 1: Identify the expression
The given expression is:
\[ \frac{10x}{5+x} \]

### Step 2: Rationalize the expression
To simplify the expression, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is \(5 - x\):
\[ \frac{10x}{5+x} \cdot \frac{5-x}{5-x} = \frac{10x(5-x)}{(5+x)(5-x)} \]

### Step 3: Expand the numerator and denominator
Now, we will expand both the numerator and the denominator:
- **Numerator**:
\[ 10x(5-x) = 50x - 10x^2 \]
- **Denominator**:
Using the difference of squares formula \(a^2 - b^2\):
\[ (5+x)(5-x) = 5^2 - x^2 = 25 - x^2 \]

### Step 4: Write the new expression
Now we can write the expression as:
\[ \frac{50x - 10x^2}{25 - x^2} \]

### Step 5: Check for common factors
Next, we check if there are any common factors in the numerator and denominator. The numerator \(50x - 10x^2\) can be factored:
\[ 10x(5 - x) \]
So now we have:
\[ \frac{10x(5 - x)}{25 - x^2} \]

### Step 6: Factor the denominator
The denominator \(25 - x^2\) can also be factored using the difference of squares:
\[ 25 - x^2 = (5 - x)(5 + x) \]

### Step 7: Simplify the expression
Now we can simplify the expression:
\[ \frac{10x(5 - x)}{(5 - x)(5 + x)} \]
We can cancel out the common factor \((5 - x)\) from the numerator and denominator (assuming \(5 - x \neq 0\)):
\[ \frac{10x}{5 + x} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{10x}{5 + x} \]