The roots of the equation $|x - 2|^2 + |x - 2| - 6 = 0$ are:

If $a, b, c$ are in Arithmetic Progression $(AP)$,then the roots of the equation $ax^2 - 2bx + c = 0$ are

Let $f(x) = x^2 + ax + b$,where $a, b \in R$. If $f(x) = 0$ has all its roots imaginary,then the roots of $f(x) + f'(x) + f''(x) = 0$ are

If the roots of the equation $(p-3)x^2 + 2(p-3)x + 2p-5 = 0$ are real and distinct for $\alpha < p < \beta$ and $(\beta - \alpha)$ is maximum,then the extreme value of the quadratic expression $-(\alpha + \beta)x^2 + \alpha \beta x + (\alpha - \beta)$ is

If $\alpha, \beta, \gamma, \delta, \varepsilon$ are the roots of the equation $x^5+x^4-13x^3-13x^2+36x+36=0$ and $\alpha < \beta < \gamma < \delta < \varepsilon$,then find the value of $\frac{\varepsilon}{\alpha}+\frac{\delta}{\beta}+\frac{1}{\gamma}$.

If $x = \sqrt[3]{{\sqrt 2 + 1}} - \sqrt[3]{{\sqrt 2 - 1}}$,then find the value of ${x^3} + 3x$.