How many roots does the equation $x - \frac{2}{x - 1} = 1 - \frac{2}{x - 1}$ have?

If the quadratic equation $3x^2 + (2k + 1)x - 5k = 0$ has real and equal roots,then the value of $k$ such that $-\frac{1}{2} < k < 0$ is

Let $S = \{ \sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0 \text{ has real roots} \}$. If $\alpha$ and $\beta$ are the smallest and largest elements of the set $S$,respectively,then $3((\alpha - 2)^2 + (\beta - 1)^2)$ equals:

The roots of the equation $x^4 - 8x^2 - 9 = 0$ are

For what values of $k$ will the equation $x^2 - 2(1 + 3k)x + 7(3 + 2k) = 0$ have equal roots?

Solve $\sqrt{5} x^{2} + x + \sqrt{5} = 0$.