Question:
If $\alpha$ and $\beta$ are the zeroes of a polynomial $f(x)=x^{2}-5 x+k$, such that $\alpha-\beta=1$, find the value of $k$.
Solution:
By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes $=\frac{-(\text { coefficient of } x)}{\text { coefficent of } x^{2}}$ and Product of zeroes $=\frac{\text { constant term }}{\text { coefficent of } x^{2}}$
$\therefore \alpha+\beta=\frac{-(-5)}{1}$ and $\alpha \beta=\frac{k}{1}$
$\Rightarrow \alpha+\beta=5$ and $\alpha \beta=\frac{k}{1}$
Solving α − β = 1 and α + β = 5, we will get
α = 3 and β = 2
Substituting these values in $\alpha \beta=\frac{k}{1}$, we will get
k = 6