Mathematics - Quadratic Equation Question with Solution | TestHub

We have, $\left|x^2-\left(7+k^2\right)x+7k^2\right|+\sqrt{(x-3)(x-3k+2)}=0$ $\Rightarrow \left|(\mathrm{x}-7)\left(\mathrm{x}-{\mathrm{k}}^2\right)\right|+\sqrt{(\mathrm{x}-3)(\mathrm{x}-(3\mathrm{x}-2))}=0$ The above equation has a solution if equations $(x-7)\left(x-k^2\right)=0$ and $(x-3)(x-(3k-2))=0$ has at least one common root. $\therefore {\mathrm{k}}^2=3\Rightarrow \mathrm{k}=\pm \sqrt{3}$ or ${\mathrm{k}}^2=3\mathrm{k}-2\Rightarrow {\mathrm{k}}^2-3\mathrm{k}+2=0$ $\Rightarrow (\mathrm{k}-1)(\mathrm{k}-2)=0\Rightarrow \mathrm{k}=1$ or 2 $3\mathrm{k}-2=7\Rightarrow \mathrm{k}=3$ ∴ Sum of all values of k is $=\sqrt{3}-\sqrt{3}+1+2+3=6$