Mathematics - Quadratic Equation Question with Solution | TestHub

Given $g(x_1)g(x_2)g(x_3)g(x_4)=A$

$\begin{array}{rl}& =(x_1^2-2)(x_3^2-2)(x_4^2-2)(x_5^2-2)\\ & =-(2-x_1^2)(2-x_2^2)(2-x_3^2)(2-x_4^2)(2-x_5^2)\\ & =2^5-\left(\sum x_1^2\right)2^4+\sum x_1^2{x}_2^2\cdot 2^3-\sum x_1^2{x}_2^2{x}_3^2\cdot 2^2\\ & +\sum x_1^2{x}_2^2{x}_3^2{x}_4^2\cdot 2-x_1^2{x}_2^2{x}_3^2{x}_4^2{x}_5^2\end{array}$

If $p(x)=x^5+x^2+1=0$ has roots $x_1,x_2,\dots ,x_5$, then the equation $q(x)$ whose roots are the squares of the roots of $p(x)$ is $q(x)=(\sqrt{y}{)}^5+(\sqrt{y}{)}^2+1=0$, where $\alpha =x$ and $y={\alpha }^2$.

$\begin{array}{rl}& \Rightarrow (y+1{)}^2=(-\sqrt{y}{)}^5\\ & \Rightarrow y^2+2y+1=y^5\\ & \Rightarrow q(x)=y^5-y^2-2y-1=0\end{array}$

Then $\sum x_1^2=\sum y_1=0$.

$\begin{array}{rl}& \sum x_1^2{x}_2^2=\sum y_1{y}_2=0\\ & \sum x_1^2{x}_2^2{x}_3^2=\sum y_1{y}_2{y}_3=1\\ & \sum x_1^2{x}_2^2{x}_3^2{x}_4^2=\sum y_1{y}_2{y}_3{y}_4=-2\\ & \sum x_1^2{x}_2^2{x}_3^2{x}_4^2{x}_5^2=\sum y_1{y}_2{y}_3{y}_4{y}_5=1\\ & ThenA=-[2^5-0+0-1\cdot 2^2-(-2)\cdot 2-1]=-[32-4+4-1]=-[31]=-31\\ & x_1{x}_2{x}_3{x}_4{x}_5=-1\Rightarrow g(x_1{x}_2\cdots x_5)=-1\\ & \Rightarrow g(x_1)g(x_2)\cdots g(x_5)-30g(x_1{x}_2\cdots x_5)=7\end{array}$