Find the numbers of sides of a polygon whose number of diagonals is
14

Answer

To find the number of sides of a polygon whose number of diagonals is 14, we can follow these steps:

### Step 1: Use the formula for the number of diagonals in a polygon.
The formula for the number of diagonals \( D \) in a polygon with \( n \) sides is given by:
\[ D = \frac{n(n - 3)}{2} \]
We know that \( D = 14 \).

### Step 2: Set up the equation.
Substituting \( D \) into the formula, we have:
\[ \frac{n(n - 3)}{2} = 14 \]

### Step 3: Eliminate the fraction.
To eliminate the fraction, we can multiply both sides of the equation by 2:
\[ n(n - 3) = 28 \]

### Step 4: Rearrange the equation.
Rearranging the equation gives us:
\[ n^2 - 3n - 28 = 0 \]

### Step 5: Factor the quadratic equation.
Now we need to factor the quadratic equation \( n^2 - 3n - 28 = 0 \). We look for two numbers that multiply to -28 and add to -3. The numbers -7 and 4 work:
\[ (n - 7)(n + 4) = 0 \]

### Step 6: Solve for \( n \).
Setting each factor to zero gives us:
1. \( n - 7 = 0 \) → \( n = 7 \)
2. \( n + 4 = 0 \) → \( n = -4 \)

### Step 7: Determine the valid solution.
Since the number of sides \( n \) cannot be negative, we discard \( n = -4 \). Thus, we have:
\[ n = 7 \]

### Conclusion:
The polygon has **7 sides**.

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