If ${b_1}{b_2} = 2({c_1} + {c_2})$,then at least one of the equations ${x^2} + {b_1}x + {c_1} = 0$ and ${x^2} + {b_2}x + {c_2} = 0$ has

If $\alpha$ and $\beta$ are the roots of $x^2 - ax + b = 0$ and if $\alpha^n + \beta^n = V_n$,then

$Quantity \, 1$: Time taken by Manoj and Shubham to complete the work together. When Manoj works alone,he takes $4.8 \, \text{hrs}$ more than the time taken by Manoj and Shubham to complete the work together. When Shubham works alone,he takes $10.8 \, \text{hrs}$ more than the time taken by both of them to complete the work together.
$Quantity \, 2$: $7.4 \, \text{hrs}$.

If $\alpha$ is a root of the quadratic equation $x^2 + 6x - 2 = 0$,then another root $\beta$ is

If $\alpha, \beta$ are the roots of $x^2 - px + q = 0$ and $\alpha', \beta'$ are the roots of $x^2 - p'x + q' = 0$,then the value of $(\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2$ is

If $x(x-3)=-1,$ then the value of $x^{3}(x^{3}-18)$ is