Question

If α and β are the zeros of the polynomial $p\left(x\right)=2x^2+5x+k$ satisfying the relation ${\mathrm{\alpha }}^2+{\mathrm{\beta }}^2+\mathrm{\alpha \beta }=\frac{21}{4}$ then find the value of k.

Answer 1

Let α and β be the zeroes of the polynomial $p\left(x\right)=2x^2+5x+k$.

$$\begin{gathered} \mathrm{Sum}\mathrm{of}\mathrm{zeroes}=-\frac{b}{a} \\ \Rightarrow \alpha +\beta =-\frac{5}{2}...\left(1\right) \\ \mathrm{and} \\ \mathrm{Product}\mathrm{of}\mathrm{zeroes}=\frac{c}{a} \\ \Rightarrow \alpha \beta =\frac{k}{2}...\left(2\right) \end{gathered}$$

Now, using (1)
$$\begin{gathered} \alpha +\beta =-\frac{5}{2} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow {\left(\alpha +\beta \right)}^2={\left(-\frac{5}{2}\right)}^2 \\ \Rightarrow {\alpha }^2+{\beta }^2+2\alpha \beta =\frac{25}{4} \\ \Rightarrow {\alpha }^2+{\beta }^2+\alpha \beta +\alpha \beta =\frac{25}{4} \\ \Rightarrow \frac{21}{4}+\alpha \beta =\frac{25}{4} \\ \Rightarrow \alpha \beta =\frac{25}{4}-\frac{21}{4} \\ \Rightarrow \alpha \beta =1 \\ \Rightarrow \frac{k}{2}=1\left(\mathrm{from}\left(2\right)\right) \\ \Rightarrow k=2 \end{gathered}$$

Hence, the value of k is 2.