Question

what is the function if Alpha beta gamma is root of equation then alpha square+beta square+gamma square

Answer 1

Answer:$\alpha ^2+\beta ^2+\gamma ^2=\frac{b^2-4c^2}{4a^2}$

Step-by-step explanation:

suppose $\alpha ,\beta ,\gamma$are the roots of

$\Rightarrow ax^3+bx^2+cx+d=0$

and we know

$(\alpha +\beta +\gamma )=-\frac{b}{2a}\quad \ldots(i)$

$\alpha \beta +\beta \gamma +\alpha \gamma =\frac{c}{a}\quad \ldots(ii)$

$\alpha \beta \gamma =-\frac{d}{a}$

squaring equation (i)  we get

$(\alpha +\beta +\gamma)^2=\frac{b^2}{4a^2}$

$\frac{b^2}{4a^2}=\alpha ^2+\beta ^2+\gamma ^2+2(\alpha \beta +\beta \gamma +\alpha \gamma )$

substitute the value of $\alpha \beta +\beta \gamma +\alpha \gamma$we get

$\frac{b^2}{4a^2}=\alpha ^2+\beta ^2+\gamma ^2+\frac{c^2}{a^2}$

$\alpha ^2+\beta ^2+\gamma ^2=\frac{b^2}{4a^2}-\frac{c^2}{a^2}$

$\alpha ^2+\beta ^2+\gamma ^2=\frac{b^2-4c^2}{4a^2}$