Solve the following quadratic equation for x;$4\sqrt{3}{x}^2 + 5x - 2\sqrt{3} = 0$

For what values of k, the roots of the quadratic equation
(k + 4) x2 + (k + 1)x + 1 = 0 are equal?

$$\begin{gathered} \mathrm{Solve} \mathrm{that} \mathrm{following} \mathrm{for} \mathrm{x}: \\ \frac{1}{2\mathrm{a}+\mathrm{b}+2\mathrm{x}} = \frac{1}{2\mathrm{a}} + \frac{1}{\mathrm{b}} + \frac{1}{2\mathrm{x}} \end{gathered}$$

Find the values of k for which the quadratic equation

9x² - 3kx + k = 0 has equal roots.

Solve for x:
   

$\frac{16}{\mathrm{x}} - 1 = \frac{15}{\mathrm{x}+1}; \mathrm{x} \ne 0, -1$

A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

Solve for x:

  $\frac{\mathrm{x}-3}{\mathrm{x}-4} + \frac{\mathrm{x}-5}{\mathrm{x}-6} = \frac{10}{3}; \mathrm{x} \ne 4, 6$

Find the value of p, for which one root of the quadratic equation px² – 14x + 8= 0 is 6 times the other.

If the roots of the equation (a² + b²) x² + 2(ac + bd) x + (c² + d² ) = 0 are
equal, prove that   $\frac{\mathrm{a}}{\mathrm{b}} = \frac{\mathrm{c}}{\mathrm{d}}$.

Solve for x:

$\frac{\mathrm{x} - 1}{2\mathrm{x} + 1} + \frac{2\mathrm{x} + 1}{\mathrm{x} - 1} = 2, \mathrm{Where} \mathrm{x}\ne -\frac{1}{2}, 1$