The three roots of equation $x^{4} - px^{3} + qx^{2} - rx + s = 0$, where $p,q,r,s \in R$ and $s &lt

Question

The three roots of equation $x^{4} - px^{3} + qx^{2} - rx + s = 0$, where $p,q,r,s \in R$ and $s < 0$, are $tanA,tanB$ and $tanC$ where $A,B,C$ are angles of a triangle. Then the fourth root of the equation can be equal to :

Accepted Answer

Let $tanD$ be the fourth root$$\begin{matrix} \Rightarrow & tan(A + B + C + D) = tan(\pi + D) = tanD = \frac{S_{1} - S_{3}}{1 - S_{2} + S_{4}} \ \Rightarrow & tanD = \frac{p - r}{1 - q + s} \ \Rightarrow & tan	ext{ }Atan	ext{ }BtanC = tanA + tanB + tanC \ \Rightarrow & \frac{	ext{ }s}{tanD} = p - tanD \ \Rightarrow & 	an^{2}D - ptanD + s = 0 \ & tanD = \frac{p \pm \sqrt{p^{2} - 4	ext{ }s}}{2} \end{matrix}$$

Question

Options

Solution

More Quadratic Equation Questions