A) \[-3\]
B) \[\frac{3}{2}\]
C) \[-\frac{\sqrt{17}}{2}\]
D) \[\frac{\sqrt{17}}{2}\]
Correct Answer: A
Solution :
Given equation are \[{{x}^{3}}+2{{x}^{2}}-5x+2=0\] ?(i) and \[{{x}^{3}}+{{x}^{2}}-8x+4=0\] ?..(ii) Now, for finding GCD of the given equations \[{{x}^{3}}+{{x}^{2}}-8x+4){{x}^{3}}+2{{x}^{2}}-5x+2(1\] \[\begin{align} & {{x}^{3}}+{{x}^{2}}-8x+4 \\ & \,\,\,--\,\,\,\,\,\,+\,\,\,\,\,\,\,- \\ & \_\_\_\_\_\_\_\_\_\_\_ \\ & {{x}^{2}}+3x-2){{x}^{3}}+{{x}^{2}}-8x+4(x-2 \\ \end{align}\] \[\begin{align} & {{x}^{2}}+3{{x}^{2}}-2x \\ & -\,\,\,\,\,-\,\,\,\,\,\,\,\,+ \\ & \_\_\_\_\_\_\_\_\_\_\_ \\ & -2{{x}^{2}}-6x+4 \\ & -2x-6x+4 \\ & +\,\,\,\,\,\,\,\,\,+\,\,\,\,\,\,\,\,\,- \\ & \_\_\_\_\_\_\_\_\_\_ \\ \end{align}\] Thus, GCD or common root of given equations is \[{{x}^{2}}+3x-2=0\] \[\therefore \] \[x=\frac{-3\pm \sqrt{{{(3)}^{2}}-4\times 1\times (-2)}}{2\times 1}\] \[\Rightarrow \] \[x=\frac{3\pm \sqrt{9+8}}{2}\] \[\Rightarrow \] \[x=\frac{-3\pm \sqrt{17}}{2}\] \[\Rightarrow \] \[x=\frac{-3+\sqrt{17}}{2},\,\,\frac{-3-\sqrt{17}}{2}\] \[\therefore \] Sum of roots \[=\frac{-3+\sqrt{17}}{2}+\frac{-3-\sqrt{17}}{2}\] \[=\frac{-6}{2}=-3\]
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