Value of an Expression Using Zeros of a Quadratic Polynomial
Video Explanation
Question
If \( \alpha \) and \( \beta \) are the zeros of the quadratic polynomial
\[ f(x) = x^2 – 5x + 4, \]
find the value of
\[ \frac{1}{\alpha} + \frac{1}{\beta} – 2\alpha\beta. \]
Solution
Step 1: Find the Zeros of the Polynomial
Factorising:
\[ x^2 – 5x + 4 = (x – 4)(x – 1) \]
So, the zeros are:
\[ \alpha = 4,\quad \beta = 1 \]
Step 2: Find Required Values
\[ \alpha + \beta = 4 + 1 = 5 \]
\[ \alpha\beta = 4 \times 1 = 4 \]
Step 3: Evaluate the Expression
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{5}{4} \]
Now,
\[ \frac{1}{\alpha} + \frac{1}{\beta} – 2\alpha\beta = \frac{5}{4} – 2(4) \]
\[ = \frac{5}{4} – 8 \]
\[ = \frac{5 – 32}{4} = -\frac{27}{4} \]
Conclusion
The required value is:
\[ \boxed{-\frac{27}{4}} \]
\[ \therefore \quad \frac{1}{\alpha} + \frac{1}{\beta} – 2\alpha\beta = -\frac{27}{4}. \]