Question

if alpha and beta are the zeroes of the polynomial x^2 - (k+6)x+2(2k-1) such that alpha + beta = 1/2 alpha* beta find the value of k

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{p}\left(\mathrm{x}\right)={\mathrm{x}}^2-\left(\mathrm{k}+6\right)\mathrm{x}+2\left(2\mathrm{k}-1\right) \\ \mathrm{Now},\mathrm{\alpha }\mathrm{and}\mathrm{\beta }\mathrm{are}\mathrm{the}\mathrm{zeroes}\mathrm{of}\mathrm{p}\left(\mathrm{x}\right),\mathrm{then} \\ \mathrm{\alpha }+\mathrm{\beta }=-\frac{\mathrm{coefficient}\mathrm{of}\mathrm{x}}{\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^2}=\frac{\mathrm{k}+6}{1}=\mathrm{k}+6 \\ \mathrm{\alpha \beta }=\frac{2\left(2\mathrm{k}-1\right)}{1}=4\mathrm{k}-2 \\ \mathrm{Now},\mathrm{\alpha }+\mathrm{\beta }=\frac{1}{2}\mathrm{\alpha \beta } \\ \Rightarrow \mathrm{k}+6=\frac{1}{2}\left(4\mathrm{k}-2\right) \\ \Rightarrow \mathrm{k}+6=2\mathrm{k}-1 \\ \Rightarrow \mathrm{k}-2\mathrm{k}=-1-6 \\ \Rightarrow -\mathrm{k}=-7 \\ \Rightarrow \boxed{\mathbf{k}\mathbf{}\mathbf{=}\mathbf{}\mathbf{7}} \end{gathered}$$