Value of an Expression Using Zeros of a Quadratic Polynomial

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeros of the quadratic polynomial

\[ p(y) = 5y^2 – 7y + 1, \]

find the value of

\[ \frac{1}{\alpha} + \frac{1}{\beta}. \]

Solution

Step 1: Use the Relationship Between Zeros and Coefficients

For a quadratic polynomial \( ay^2 + by + c \):

\[ \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a} \]

Here,

\[ a = 5,\quad b = -7,\quad c = 1 \]

\[ \alpha + \beta = -\frac{-7}{5} = \frac{7}{5} \]

\[ \alpha\beta = \frac{1}{5} \]

Step 2: Evaluate the Required Expression

\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} \]

Substitute the values:

\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\frac{7}{5}}{\frac{1}{5}} \]

\[ = 7 \]

Conclusion

The required value is:

\[ \boxed{7} \]

\[ \therefore \quad \frac{1}{\alpha} + \frac{1}{\beta} = 7. \]