Let $\alpha, \beta$ be the roots of $x^2 + (3 - \lambda)x - \lambda = 0$. The value of $\lambda$ for which $\alpha^2 + \beta^2$ is minimum,is

The minimum value of the sum of the squares of the roots of $x^{2}+(3-a)x+1=2a$ is:

If one root of the equation $x^3-9x^2+26x-24=0$ is twice the other,then the sum of the cubes of those two roots is

If the harmonic mean of the roots of the equation $\sqrt{2} x^2 - bx + (8 - 2\sqrt{5}) = 0$ is $4$,then the value of $b$ is

If $a, -a, b$ are the roots of $x^{3}-5x^{2}-x+5=0$,then $b$ is a root of

Let $a, b, c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a, c), (2, b)$ and $(a, b)$ be $\left(\frac{10}{3}, \frac{7}{3}\right)$. If $\alpha, \beta$ are the roots of the equation $ax^{2} + bx + 1 = 0$,then the value of $\alpha^{2} + \beta^{2} - \alpha\beta$ is ....... .