$x = \dots$ is a solution of the quadratic equation $x^{2} + 7x + 12 = 0$.

Values of $k$ for which the quadratic equation $2x^{2}-kx+k=0$ has equal roots is

$A$ two-digit number is such that the product of its digits is $10$. When $27$ is added to the original number,the new number obtained is the same as the number obtained by interchanging the digits of the original number. Determine the number.

Find the roots of the following quadratic equation by the method of completing the square: $x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$

The faster train takes $1$ hour less than the slower train for covering a $400\, km$ distance. If the speed of the slower train is $20\, km/hr$ less than the speed of the faster train,find the speed of the slower train.

Verify whether the given value of $x$ is a solution of the quadratic equation or not: $x + \frac{1}{x} = 3 \frac{1}{3}$; $x = \frac{1}{3}$.