Solve for `x: x(x - (5x + 6)/(x)) = 0`

Answer

To solve the equation \( x\left(x - \frac{5x + 6}{x}\right) = 0 \), we will follow these steps:

### Step 1: Simplify the Equation
Start with the given equation:
\[ x\left(x - \frac{5x + 6}{x}\right) = 0 \]
We can simplify the term inside the parentheses:
\[ x - \frac{5x + 6}{x} = x - (5 + \frac{6}{x}) = x - 5 - \frac{6}{x} \]
Thus, the equation becomes:
\[ x\left(x - 5 - \frac{6}{x}\right) = 0 \]

### Step 2: Eliminate the Fraction
To eliminate the fraction, multiply the entire equation by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 - 5x - 6 = 0 \]

### Step 3: Factor the Quadratic Equation
Next, we will factor the quadratic equation \( x^2 - 5x - 6 = 0 \). We need to find two numbers that multiply to \(-6\) (the constant term) and add to \(-5\) (the coefficient of \( x \)):
The numbers \(-6\) and \(1\) satisfy these conditions:
\[ (x - 6)(x + 1) = 0 \]

### Step 4: Set Each Factor to Zero
Now we set each factor equal to zero:
1. \( x - 6 = 0 \)
2. \( x + 1 = 0 \)

### Step 5: Solve for \( x \)
Solving these equations gives:
1. \( x = 6 \)
2. \( x = -1 \)

### Final Solution
The solutions to the equation \( x\left(x - \frac{5x + 6}{x}\right) = 0 \) are:
\[ x = 6 \quad \text{or} \quad x = -1 \]

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