Question

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

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

Answer 1

Answer: (A)

Given equation are $x^3+2x^2-5x+2=0$ ?(i) and $x^3+x^2-8x+4=0$ ..(ii)
Now, for finding GCD of the given equations $x^3+x^2-8x+4)x^3+2x^2-5x+2(1$ $\begin{array}{rl}& x^3+x^2-8x+4\\ & --+-\\ & \_\_\_\_\_\_\_\_\_\_\_\\ & x^2+3x-2)x^3+x^2-8x+4(x-2\end{array}$ $\begin{array}{rl}& x^2+3x^2-2x\\ & --+\\ & \_\_\_\_\_\_\_\_\_\_\_\\ & -2x^2-6x+4\\ & -2x-6x+4\\ & ++-\\ & \_\_\_\_\_\_\_\_\_\_\end{array}$ 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 $=\frac{-3+\sqrt{17}}{2}+\frac{-3-\sqrt{17}}{2}$ $=\frac{-6}{2}=-3$