Discuss the nature of the roots of the following equations :

(i) $3x^2 - 7x + 8 = 0$

(ii) $x^2 - \dfrac{1}{2}x - 4 = 0$

(iii) $5x^2 - 6\sqrt{5}x + 9 = 0$

(iv) $\sqrt{3}x^2 - 2x - \sqrt{3} = 0$

In case real roots exist , then find them.

Find the values of k so that the quadratic equation (4 - k)x² + 2(k + 2)x + (8k + 1) = 0 has equal roots.

Find the value(s) of k for which each of the following quadratic equation has equal roots:

(i) 3kx² = 4(kx - 1)

(ii) (k + 4)x² + (k + 1)x + 1 = 0

Also, find the roots for that value(s) of k in each case.

Find two natural numbers which differ by 3 and whose squares have the sum 117.