Question

The sum of the common roots of the equations , ${{x}^{3}}+2{{x}^{2}}-5x+2=0~$ and ${{x}^{3}}+\text{ }{{x}^{2}}-8x+4=0,$ is

• A. $-3$
• B. $\frac{3}{2}$
• C. $-\frac{\sqrt{17}}{2}$
• D. $\frac{\sqrt{17}}{2}$

Answer 1

Answer: (A)

$-3$

Answer 2

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{aligned} & {{x}^{3}}+{{x}^{2}}-8x+4 \\\ & \,\,\,--\,\,\,\,\,\,+\,\,\,\,\,\,\,- \\\ & \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ \\\ & {{x}^{2}}+3x-2){{x}^{3}}+{{x}^{2}}-8x+4(x-2 \\\ \end{aligned} \begin{aligned} & {{x}^{2}}+3{{x}^{2}}-2x \\\ & -\,\,\,\,\,-\,\,\,\,\,\,\,\,+ \\\ & \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ \\\ & -2{{x}^{2}}-6x+4 \\\ & -2x-6x+4 \\\ & +\,\,\,\,\,\,\,\,\,+\,\,\,\,\,\,\,\,\,- \\\ & \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ \\\ \end{aligned} 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 $=\frac{-3+\sqrt{17}}{2}+\frac{-3-\sqrt{17}}{2}$ $=\frac{-6}{2}=-3$