Question

Find the value of m if one zero of the polynomial is (m²⁺4)x²+63x+4m is reciprocal of the other .(Hint:product of zero =1)

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{p}\left(\mathrm{x}\right)=\left({\mathrm{m}}^2+4\right){\mathrm{x}}^2+63\mathrm{x}+4\mathrm{m} \\ \mathrm{Let}\mathrm{\alpha }\mathrm{and}\mathrm{\beta }\mathrm{are}\mathrm{zeroes}\mathrm{of}\mathrm{p}\left(\mathrm{x}\right). \\ \mathrm{Now},\mathrm{\alpha \beta }=\frac{\mathrm{constant}\mathrm{term}}{\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^2}=\frac{4\mathrm{m}}{\left({\mathrm{m}}^2+4\right)} \\ \mathrm{Now},\mathrm{\beta }=\frac{1}{\mathrm{\alpha }} \\ \mathrm{So},\mathrm{\alpha \beta }=\frac{4\mathrm{m}}{\left({\mathrm{m}}^2+4\right)} \\ \Rightarrow \mathrm{\alpha }\times \frac{1}{\mathrm{\alpha }}=\frac{4\mathrm{m}}{\left({\mathrm{m}}^2+4\right)} \\ \Rightarrow 1=\frac{4\mathrm{m}}{\left({\mathrm{m}}^2+4\right)} \\ \Rightarrow {\mathrm{m}}^2-4\mathrm{m}+4=0 \\ \Rightarrow {\mathrm{m}}^2-2\mathrm{m}-2\mathrm{m}+4=0 \\ \Rightarrow \mathrm{m}\left(\mathrm{m}-2\right)-2\left(\mathrm{m}-2\right)=0 \\ \Rightarrow \left(\mathrm{m}-2\right)\left(\mathrm{m}-2\right)=0 \\ \Rightarrow \mathrm{m}-2=0\mathrm{or}\mathrm{m}-2=0 \\ \Rightarrow \mathrm{m}=2\mathrm{or}\mathrm{m}=2 \\ \mathrm{So},\mathbf{m}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2} \end{gathered}$$