উদাহরণ ৫৷ [tex]{x^2} + 3x + 4 = 0[/tex] এই সমীকরণের বীজ দুটি [tex]\alpha ,\beta [/tex] হলে, যে সমীকরণের বীজ দুটি [tex]{\left( {\alpha + \beta } \right)^2},{\left( {\alpha - \beta } \right)^2}[/tex] তা নির্ণয় করো। [H.S ‘84]
সমাধান: [tex]{x^2} + 3x + 4 = 0[/tex] এই সমীকরণের বীজ দুটি হল [tex]\alpha ,\beta [/tex]।
অতএব [tex]\alpha + \beta = - 3 \to \left( 1 \right),\alpha \beta = 4 \to \left( 2 \right)[/tex]
সুতরাং [tex]{\left( {\alpha + \beta } \right)^2} = {\left( { - 3} \right)^2} = 9 \to \left( 3 \right)[/tex]
আমরা জানি
[tex]\begin{array}{l}
{\left( {\alpha - \beta } \right)^2}\\
= {\left( {\alpha + \beta } \right)^2} - 4\alpha \beta \left[ {by\left( 1 \right),\left( 2 \right)} \right]\\
= {\left( { - 3} \right)^2} - 4 \times 4\\
= 9 - 16\\
= - 7\\
\Rightarrow {\left( {\alpha - \beta } \right)^2} = - 7 \to \left( 4 \right)
\end{array}[/tex]
(3),(4) ব্যবহার করে পাই
[tex]\begin{array}{l}
{\left( {\alpha + \beta } \right)^2} + {\left( {\alpha - \beta } \right)^2} = 9 - 7 = 2 \to \left( 5 \right)\\
{\left( {\alpha + \beta } \right)^2} \times {\left( {\alpha - \beta } \right)^2} = 9 \times \left( { - 7} \right) = - 63 \to \left( 6 \right)
\end{array}[/tex]
(5) ও (6) ব্যবহার করে পাই
[tex]\begin{array}{l}
{x^2} - \left\{ {{{\left( {\alpha + \beta } \right)}^2} + {{\left( {\alpha - \beta } \right)}^2}} \right\}x + {\left( {\alpha + \beta } \right)^2}{\left( {\alpha - \beta } \right)^2} = 0\\
\Rightarrow {x^2} - 2x - 63 = 0
\end{array}[/tex]