Value of an Expression Using Zeros of a Quadratic Polynomial

Video Explanation

Question

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

\[ f(x) = x^2 – x – 4, \]

find the value of

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

Solution

Step 1: Use Relations Between Zeros and Coefficients

For a quadratic polynomial \( ax^2 + bx + c \),

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

Here,

\[ a = 1,\quad b = -1,\quad c = -4 \]

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

\[ \alpha\beta = \frac{-4}{1} = -4 \]

Step 2: Find \( \dfrac{1}{\alpha} + \dfrac{1}{\beta} \)

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

\[ = \frac{1}{-4} = -\frac{1}{4} \]

Step 3: Evaluate the Required Expression

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

\[ = -\frac{1}{4} – (-4) \]

\[ = -\frac{1}{4} + 4 \]

\[ = \frac{15}{4} \]

Conclusion

The required value is:

\[ \boxed{\frac{15}{4}} \]

\[ \therefore \quad \frac{1}{\alpha} + \frac{1}{\beta} – \alpha\beta = \frac{15}{4}. \]