Question

If α and β are the zeroes of a polynomial f(x) = x² − 5x + k, such that α − β = 1, find the value of k.

Answer 1

By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = $\frac{-\left(\mathrm{coefficient}\mathrm{of}\mathit{}x\right)}{\mathrm{coefficent}\mathrm{of}{x}^2}$ and Product of zeroes = $\frac{\mathrm{constant}\mathrm{term}}{\mathrm{coefficent}\mathrm{of}{x}^2}$
$$\begin{gathered} \therefore \alpha +\beta =\frac{-\left(-5\right)}{1}\mathrm{and}\alpha \beta =\frac{k}{1} \\ \Rightarrow \alpha +\beta =5\mathrm{and}\alpha \beta =\frac{k}{1} \end{gathered}$$
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