The value of such that the sum of the squares of the roots of x^2+(3-)x+2= assumes the lea…

Question

The value of $\lambda$ such that the sum of the squares of the roots of $x^2+(3-\lambda)x+2=\lambda$ assumes the least value is:

Accepted Answer

$\alpha+\beta=\lambda-3$, $\alpha\beta=2-\lambda$.

$$\alpha^2+\beta^2=(\lambda-3)^2-2(2-\lambda)=\lambda^2-4\lambda+5=(\lambda-2)^2+1,$$

minimized at $\lambda=2.$

Solutions · 1