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

The Correct Option is A

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 $