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In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1] In terms of a new quantity x − h {\displaystyle x-h} , this expression is a quadratic polynomial with no linear term.
To complete the square, form a squared binomial on the left-hand side of a quadratic equation, from which the solution can be found by taking the square root of both sides. The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation a x 2 + b x + c = 0 {\displaystyle ...
In three-dimensional hyperbolic geometry, the alternated order-6 hexagonal tiling honeycomb is a uniform compact space-filling tessellation (or honeycomb).As an alternation, with Schläfli symbol h{4,3,6} and Coxeter-Dynkin diagram or , it can be considered a quasiregular honeycomb, alternating triangular tilings and tetrahedra around each vertex in a trihexagonal tiling vertex figure.
Points with suffix P are in the Z plane and points with suffix Q are in the Y plane. Therefore, transformations P 1 to Q 1 and P 3 to Q 3 are from the Z Smith chart to the Y Smith chart and transformation Q 2 to P 2 is from the Y Smith chart to the Z Smith chart. The following table shows the steps taken to work through the remaining components ...
Solutions (not necessarily optimal) have been computed for every N ≤ 10,000. [2] Solutions up to N = 20 are shown below. [2] The obvious square packing is optimal for 1, 4, 9, 16, 25, and 36 circles (the six smallest square numbers), but ceases to be optimal for larger squares from 49 onwards.
In Coxeter notation can be represented as [6,4 *], removing two of three mirrors (passing through the square center) in the [6,4] symmetry. The *3333 symmetry can be doubled to 663 symmetry by adding a mirror bisecting the fundamental domain. This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry.
The coefficient a is called the slope of the function and of the line (see below). If the slope is a = 0 {\displaystyle a=0} , this is a constant function f ( x ) = b {\displaystyle f(x)=b} defining a horizontal line, which some authors exclude from the class of linear functions. [ 3 ]
The next step is to insert a variable y into the perfect square on the left side of equation , and a corresponding 2y into the coefficient of u 2 in the right side. To accomplish these insertions, the following valid formulas will be added to equation ( 2 ),