If $64$ is the discriminant of $k x^{2}-4 x-4=0$,then the value of $k$ is ...........

Solve the following equation using the method of factorization: $(x+1)^{2} + x^{2} = 221$.

If the value of the discriminant of the quadratic equation $x^{2}-10x+(2k-1)=0$ is $40$,then $k = \ldots$.

The product of the digits of a two-digit number is $14$. The number obtained by interchanging the digits is $45$ more than the original number. Find the original number.

Solve the following quadratic equation using the quadratic formula,if the equation has a solution in $R$: $48m^{2} - 13m - 1 = 0$.

If $x=\sqrt{2}$ is one of the roots of the equation $ax^{2}+\sqrt{2}bx+2c=0$; $a \neq 0$,$a, b, c \in R$,then prove that $a+b+c=0$.