Question

If -5 is a root of the quadratic equation 2x^2+px-15=0 and the quadratic equation p(x^2+x)k=0 has qual root, find the value of k

Answer 1

Dear student
$$\begin{gathered} \mathrm{Given}:-5\mathrm{is}\mathrm{a}\mathrm{root}\mathrm{of}2{\mathrm{x}}^2+\mathrm{px}-15=0 \\ \Rightarrow 2{\left(-5\right)}^2+\mathrm{p}\left(-5\right)-15=0 \\ \Rightarrow 50-5\mathrm{p}-15=0 \\ \Rightarrow 35=5\mathrm{p} \\ \Rightarrow \mathrm{p}=7 \\ \mathrm{Consider}\mathrm{p}\left({\mathrm{x}}^2+\mathrm{x}\right)\mathrm{k}=0 \\ \Rightarrow 7\left({\mathrm{x}}^2+\mathrm{x}\right)\mathrm{k}=0 \\ \Rightarrow \left(7{\mathrm{x}}^2+7\mathrm{x}\right)\mathrm{k}=0 \\ \Rightarrow 7{\mathrm{kx}}^2+7\mathrm{kx}=0 \\ \mathrm{Since}\mathrm{equation}\mathrm{has}\mathrm{equal}\mathrm{roots}\mathrm{so}\mathrm{discrminant}\mathrm{will}\mathrm{be}0. \\ \mathrm{Here}\mathrm{on}\mathrm{comparing}\mathrm{with}\mathrm{general}\mathrm{equation}{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0 \\ \mathrm{a}=7\mathrm{k},\mathrm{b}=7\mathrm{k},\mathrm{c}=0 \\ \mathrm{So},\mathrm{D}={\mathrm{b}}^2-4\mathrm{ac}=0 \\ \Rightarrow {\left(7\mathrm{k}\right)}^2-4\left(7\mathrm{k}\right)\left(0\right)=0 \\ \Rightarrow 49{\mathrm{k}}^2=0 \\ \Rightarrow \mathrm{k}=0 \end{gathered}$$
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