Solve each of the following quadratic equations:
`(16)/(x)-1=(15)/(x+1),xne0,-1`

Answer

To solve the quadratic equation \( \frac{16}{x} - 1 = \frac{15}{x + 1} \), we will follow these steps:

### Step 1: Eliminate the fractions
We start by eliminating the fractions by finding a common denominator. The common denominator for \( x \) and \( x + 1 \) is \( x(x + 1) \). We will multiply both sides of the equation by this common denominator.

\[ x(x + 1) \left( \frac{16}{x} - 1 \right) = x(x + 1) \left( \frac{15}{x + 1} \right) \]

### Step 2: Distribute and simplify
Distributing on both sides:

\[ 16(x + 1) - x(x + 1) = 15x \]

This simplifies to:

\[ 16x + 16 - (x^2 + x) = 15x \]

### Step 3: Rearrange the equation
Now, we will rearrange the equation to bring all terms to one side:

\[ 16x + 16 - x^2 - x - 15x = 0 \]

Combining like terms gives:

\[ -x^2 + 16 - x = 0 \]

### Step 4: Multiply through by -1
To make the leading coefficient positive, we multiply the entire equation by -1:

\[ x^2 - 16 + x = 0 \]

Rearranging gives:

\[ -x^2 + x + 16 = 0 \]

### Step 5: Factor the quadratic
We can factor the quadratic equation:

\[ x^2 - 16 = 0 \]

This can be factored as:

\[ (x - 4)(x + 4) = 0 \]

### Step 6: Solve for x
Setting each factor equal to zero gives us:

\[ x - 4 = 0 \quad \text{or} \quad x + 4 = 0 \]

Thus, we find:

\[ x = 4 \quad \text{or} \quad x = -4 \]

### Step 7: Check for restrictions
We need to ensure that our solutions do not violate the restrictions \( x \neq 0 \) and \( x \neq -1 \). Both \( x = 4 \) and \( x = -4 \) are valid solutions.

### Final Answer
The solutions to the equation are:

\[ x = 4 \quad \text{and} \quad x = -4 \]