# 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) = 3 ^{2} = 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.

For example, **364 ^{2} = 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.

**(3x ^{2} + 4x + 2)^{2} = 9x^{4} + 24x^{3} + 28x^{2} + 16x + 4**