A polygon has 35 diagonals. Find the number of sides.

Answer

To find the number of sides of a polygon that has 35 diagonals, we can use the formula for the number of diagonals in a polygon:

\[ D = \frac{n(n - 3)}{2} \]

where \(D\) is the number of diagonals and \(n\) is the number of sides.

### Step 1: Set up the equation
Given that the number of diagonals \(D = 35\), we can set up the equation:

\[ \frac{n(n - 3)}{2} = 35 \]

### Step 2: Multiply both sides by 2
To eliminate the fraction, multiply both sides of the equation by 2:

\[ n(n - 3) = 70 \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:

\[ n^2 - 3n - 70 = 0 \]

### Step 4: Factor the quadratic equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to -70 and add to -3. These numbers are -10 and 7. Thus, we can write:

\[ (n - 10)(n + 7) = 0 \]

### Step 5: Solve for \(n\)
Setting each factor equal to zero gives us:

1. \(n - 10 = 0 \Rightarrow n = 10\)
2. \(n + 7 = 0 \Rightarrow n = -7\)

Since the number of sides cannot be negative, we discard \(n = -7\).

### Conclusion
Thus, the number of sides \(n\) of the polygon is:

\[ n = 10 \]