Question

Find the value of k for the following quadratic equation, when the roots are equal:
(k - 4)x^​2 + 2(k - 4)x + 4 = 0

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{quadratic}\mathrm{equation}\mathrm{is}: \\ \left(\mathrm{k}-4\right){\mathrm{x}}^2+2\left(\mathrm{k}-4\right)\mathrm{x}+4=0 \\ \mathrm{Now},\mathrm{a}=\left(\mathrm{k}-4\right);\mathrm{b}=2\left(\mathrm{k}-4\right);\mathrm{c}=4 \\ \mathrm{Now},\mathrm{D}={\mathrm{b}}^2-4\mathrm{ac} \\ =4{\left(\mathrm{k}-4\right)}^2-4\left(\mathrm{k}-4\right)\left(4\right) \\ =4\left({\mathrm{k}}^2+16-8\mathrm{k}\right)-16\mathrm{k}+64 \\ =4{\mathrm{k}}^2+128-48\mathrm{k} \\ \mathrm{For}\mathrm{equal}\mathrm{roots}: \\ \mathrm{D}=0 \\ \Rightarrow 4{\mathrm{k}}^2-48\mathrm{k}+128=0 \\ \Rightarrow {\mathrm{k}}^2-12\mathrm{k}+32=0 \\ \Rightarrow {\mathrm{k}}^2-8\mathrm{k}-4\mathrm{k}+32=0 \\ \Rightarrow \mathrm{k}\left(\mathrm{k}-8\right)-4\left(\mathrm{k}-8\right)=0 \\ \Rightarrow \left(\mathrm{k}-4\right)\left(\mathrm{k}-8\right)=0 \\ \Rightarrow \mathrm{k}-4=0\mathrm{or}\mathrm{k}-8=0 \\ \Rightarrow \mathrm{k}=4\mathrm{or}\mathrm{k}=8 \end{gathered}$$