If $\alpha, \beta$ and $\gamma$ are the roots of the equation $2x^3 - 3x^2 + 6x + 1 = 0$,then $\alpha^2 + \beta^2 + \gamma^2$ is equal to

If $\alpha$ and $\beta$ are the real roots of the equation $x^2+ax+b=0$,where $\alpha+\beta=\frac{1}{2}$ and $\alpha^3+\beta^3=\frac{37}{8}$,then find the value of $a-\frac{1}{b}$.

If the arithmetic mean and harmonic mean of the roots of a quadratic equation are $3/2$ and $4/3$ respectively,then what is the equation?

Let $\alpha, \beta, \gamma$ be the roots of $x^3+x+10=0$ and $\alpha_1=\frac{\alpha+\beta}{\gamma^2}, \beta_1=\frac{\beta+\gamma}{\alpha^2}, \gamma_1=\frac{\gamma+\alpha}{\beta^2}$. Then,the value of $(\alpha_1^3+\beta_1^3+\gamma_1^3)-\frac{1}{10}(\alpha_1^2+\beta_1^2+\gamma_1^2)$ is

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 ....... .

If the sum of any two roots of the equation $x^3+p x^2+q x+r=0$ is zero,then