# MSC #26 – The Duplex and its Extensions.

The Duplex is an important devise in squaring numbers and polynomials, extracting square roots and solving quadratic equations. Its principles can be extended into triplex and beyond.

The duplex of a number with an **even**
number of digits is twice the sum of the product of the symmetrically placed
digits.

D(ab) = 2ab

D(46) = 2 (4 x 6) = 48

D(abcd) = 2 x (ad + bc)

D(2345) = 2 x [(2 x 5) + (3 x 4)] = 44

The duplex of a number with an odd number of digits is square of the middle digit plus twice the sum of the products of to the symmetrically placed digits.

D(3) = 9

D(345) = 4^{2} + 2 ( 3 x 5)
= 16 + 30 = 46

D(abcde) = c^{2} + 2 x [(a x
e) + (b x d)]

D(76543) = 5^{2} + 2 x [(7 x
3) + (6 x 4)] = 25 + 2(45) = 115

The square of a number is the sum of its duplexes.

In our example, 364^{2} =
D(3) + D(36) + D(364) + D(64) + D(4)

= 9/36/60/48/16 = 132,496

The square of (3x^{2} + 4x + 1) = 9x^{4} + 24x^{3} + 22x^{2} + 8x + 1

Triplexes, square and cube roots and other applications are discussed in Chapter 26 of “30 Master Strategies in Computing.”