In Hindu civilization,$\ldots \ldots \ldots$ gave the method to solve quadratic equations.

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

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.

The length of the hypotenuse of an isosceles right-angled triangle is $8 \, cm$. Find the length of the other two sides.

Find the roots of the following quadratic equation by the method of completing the square: $2x^{2} - 7x + 3 = 0$.

If one of the roots of the equation $x^{2}-(1+k)x+\sqrt{2}=0$ is $\sqrt{2},$ then find $k$.