MSC # 26 – Duplex

The Duplex is an important device 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 products 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) = 3² = 9

D(345) = 4² + 2 ( 3 x 5) = 16 + 30 = 46

D(abcde) = c² + 2 x [(a x e) + (b x d)]

D(76543) = 5² + 2 x [(7 x 3) + (6 x 4)] = 25 + 2(45) = 115

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

For example, 364² = D(3) + D(36) + D(364) + D(64) + D(4)

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

Squaring polynomials can be done in one line using duplexes.

(3x² + 4x + 2)²  = 9x⁴ + 24x³ + 28x² + 16x + 4