Form the values of `(1)/(alpha)+(1)/(beta)` in terms of a, b, c if `alpha, beta` are roots of `ax^(2)+bx+c=0`, `c != 0`

Answer

To find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) in terms of \( a, b, c \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Use the formula for the sum of the roots
The sum of the roots \( \alpha + \beta \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]

### Step 2: Use the formula for the product of the roots
The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} \]

### Step 3: Express \( \frac{1}{\alpha} + \frac{1}{\beta} \)
We can express \( \frac{1}{\alpha} + \frac{1}{\beta} \) using the sum and product of the roots:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

### Step 4: Substitute the values from steps 1 and 2
Now, substituting the values from Step 1 and Step 2 into the equation:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{b}{a}}{\frac{c}{a}} \]

### Step 5: Simplify the expression
When we simplify this expression, we get:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-b}{c} \]

### Final Answer
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) in terms of \( a, b, c \) is:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = -\frac{b}{c} \]