The number of diagonals of a convex polygon is 15 less than 4 times the number of its sides. The number of sides of the polygon be

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem, we need to find the number of sides \( n \) of a convex polygon given that the number of diagonals is 15 less than 4 times the number of its sides.

### Step-by-Step Solution:

1. **Understanding the Formula for Diagonals**:
The formula for the number of diagonals \( D \) in a convex polygon with \( n \) sides is given by:
\[ D = \frac{n(n - 3)}{2} \]

2. **Setting Up the Equation**:
According to the problem, the number of diagonals is also described as:
\[ D = 4n - 15 \]
Therefore, we can set the two expressions for \( D \) equal to each other:
\[ \frac{n(n - 3)}{2} = 4n - 15 \]

3. **Eliminating the Fraction**:
To eliminate the fraction, multiply both sides of the equation by 2:
\[ n(n - 3) = 2(4n - 15) \]
This simplifies to:
\[ n(n - 3) = 8n - 30 \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ n^2 - 3n = 8n - 30 \]
Moving all terms to one side results in:
\[ n^2 - 3n - 8n + 30 = 0 \]
Simplifying further:
\[ n^2 - 11n + 30 = 0 \]

5. **Factoring the Quadratic Equation**:
Now we need to factor the quadratic equation:
\[ n^2 - 11n + 30 = 0 \]
We look for two numbers that multiply to 30 and add to -11. These numbers are -6 and -5:
\[ (n - 6)(n - 5) = 0 \]

6. **Finding the Solutions**:
Setting each factor to zero gives:
\[ n - 6 = 0 \quad \text{or} \quad n - 5 = 0 \]
Therefore, the solutions are:
\[ n = 6 \quad \text{or} \quad n = 5 \]

7. **Conclusion**:
The number of sides of the polygon can be either 5 or 6.

### Final Answer:
The number of sides of the polygon is \( n = 5 \) or \( n = 6 \).