If α and β are the zeroes of a polynomial f(x)


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 $k$.



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


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