In the diagram, ABDE is a rectangle, ABCD is equilateral, and AD is parallel to BC. Also, AE = 2x for some real number x. (i) Determine the length of AB in terms of x. (ii) Determine positive integers r and s for which AC/AD =square root r/s (a) Suppose that a = 5 and b = 4. Determine all pairs of integers (K, L) for which K2 + 3L2= a2 + b2 − ab. (b) Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 + 3L2 = a2 + b2 − ab. (c) Prove that, for all integers a and b, there is at least one pair of integers (K, L) for which K2 + 3L2 = a2 + b2 − ab.

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Roots

The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.