Find the numbers of sides of a polygon whose number of diagonals is
5

Answer

To find the number of sides of a polygon whose number of diagonals is 5, 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 the number of diagonals \( D \) is 5, so we can set up the equation:
\[ 5 = \frac{n(n - 3)}{2} \]

### Step 2: Eliminate the fraction by multiplying both sides by 2
To eliminate the fraction, we multiply both sides of the equation by 2:
\[ 2 \times 5 = n(n - 3) \]
This simplifies to:
\[ 10 = n(n - 3) \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ n(n - 3) - 10 = 0 \]
or
\[ n^2 - 3n - 10 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation \( n^2 - 3n - 10 = 0 \). We look for two numbers that multiply to -10 and add to -3. The numbers -5 and 2 work:
\[ (n - 5)(n + 2) = 0 \]

### Step 5: Solve for \( n \)
Now, we set each factor equal to zero:
1. \( n - 5 = 0 \) gives \( n = 5 \)
2. \( n + 2 = 0 \) gives \( n = -2 \)

Since the number of sides \( n \) cannot be negative, we discard \( n = -2 \).

### Conclusion
Thus, the number of sides of the polygon is:
\[ n = 5 \]