If the sum and product of the zeros of the polynomial `ax^2 – 5x + c` is 10 find a and c.

Answer

To solve the problem, we need to find the values of \( a \) and \( c \) given that the sum and product of the zeros of the polynomial \( ax^2 - 5x + c \) is 10.

### Step-by-Step Solution:

1. **Identify the Polynomial**: The given polynomial is \( ax^2 - 5x + c \).

2. **Sum and Product of Zeros**: Let the zeros of the polynomial be \( \alpha \) and \( \beta \). We know:
- Sum of the zeros \( \alpha + \beta = 10 \)
- Product of the zeros \( \alpha \beta = 10 \)

3. **Using Vieta's Formulas**: According to Vieta's formulas for a quadratic polynomial \( ax^2 + bx + c \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

For our polynomial \( ax^2 - 5x + c \):
- Here, \( b = -5 \) and \( c = c \).

4. **Set Up the Equations**:
- From the sum of the roots:
\[ \alpha + \beta = -\frac{-5}{a} = \frac{5}{a} \]
Setting this equal to 10:
\[ \frac{5}{a} = 10 \]
- From the product of the roots:
\[ \alpha \beta = \frac{c}{a} \]
Setting this equal to 10:
\[ \frac{c}{a} = 10 \]

5. **Solve for \( a \)**:
- From the equation \( \frac{5}{a} = 10 \):
\[ 5 = 10a \implies a = \frac{5}{10} = \frac{1}{2} \]

6. **Solve for \( c \)**:
- Substitute \( a \) into the equation \( \frac{c}{a} = 10 \):
\[ \frac{c}{\frac{1}{2}} = 10 \implies c = 10 \times \frac{1}{2} = 5 \]

7. **Final Values**:
- Thus, we have \( a = \frac{1}{2} \) and \( c = 5 \).

### Summary of Results:
- \( a = \frac{1}{2} \)
- \( c = 5 \)