Mathematics - Quadratic Equation Question with Solution | TestHub

We have, $x^2+(3-\lambda )x+2=\lambda , \text{have roots }\alpha \& \beta .$

Then, $f\left(\lambda \right)={\alpha }^2+{\beta }^2$

$={\left(\alpha +\beta \right)}^2-2\alpha \beta ={\left(3-\lambda \right)}^2-2\left(2-\lambda \right)={\lambda }^2-6\lambda +9-4+2\lambda ={\lambda }^2-4\lambda +5$

$f^{\prime }\left(\lambda \right)=2\lambda -4=0$ $\{$ for the minimum value of $f\left(\lambda \right)\}$

As $f^{\text{'}\text{'}}\left(\lambda \right)=2>0$

$\Rightarrow \lambda =2$