Instruction

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If you want to write

**the number**in**the square**with an editor that does not support text formatting, it is best to use programmers invented a way to denote powers of "circumflex". This icon is in between**number**m and its degree and first appeared in the BASIC language. Before him, there were other options, but they are not spread enough. And this sign is now often used to denote the degree of outside computer. Keyboard the circumflex is entered by pressing the SHIFT and 6 must be enabled English keyboard layout. Looks**the number**in**the square**using the circumflex, for example:1586^22

The other method applies to editors who know how to shift the baseline of individual letters and numbers up or down in relation to adjacent signs. This gives you the ability to use the usual "Cartesian" designation of the degree. Usually characters with this bias is called the "upper (or lower) index", and sometimes "Superscript (or subscript) sign". For example, in a text editor Microsoft Word to write the same

**number**1586 in**the square**, first dial 15862, then select last pair and click the icon with the x in**the square**. It is placed in the section "Font" section of the "Home" menu of the editor.3

If you want to write

**the number**in degrees in the source code of a web document, use the command that tells the browser that the letter which denotes the degree must be shifted up relative to the baseline the rest of the text. Such command on the language HTML (HyperText Markup Language "hypertext markup language") are called "tags". You need a tag consists of the opening<sup>) and closing (</sup>) of the parts between which is placed the figure indicating the degree number. For example, this HTML snippet of the page might look like this:1586<sup>2</sup># Advice 2: How to be a square of binomials

The method of allocation of square binomials used in simplifying the cumbersome expressions, and solving quadratic equations. In practice it is usually combined with other techniques, including factorization, grouping, etc.

Instruction

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The method of singling out a complete square of binomials is based on the use of two formulas of the reduced multiplication of polynomials. These formulas are special cases of the Binomial theorem for the second degree and allow you to simplify the search expression so that it was possible to conduct further reduction or decomposition on the multipliers:

(m + n)2 = m2 + 2·m·n + n2;

(m - n)2 = m2 - 2·m·n + n2.

(m + n)2 = m2 + 2·m·n + n2;

(m - n)2 = m2 - 2·m·n + n2.

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According to this method from the original polynomial is required to allocate the squares of the two monomials and the sum/difference of their double works. The application of this method makes sense if the high-degree terms not less than 2. Suppose the task is to factorize a lowering of the degree the following expression:

4·y^4 + z^4

4·y^4 + z^4

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To solve the problem you need to use the method of allocation of the full square. Thus, the term consists of two variables with monomials of even degree. Therefore, it is possible to identify each of them using m and n:

m = 2·y2; n = z2.

m = 2·y2; n = z2.

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Now we need to give initial expression to the form (m + n)2. It already contains the squares of these terms, but not enough double work. You need to add it artificially, and then subtract:

(2·y2)2 + 2·2·y2·z2 + (z2)2 - 2·2·y2 ·z2 = (2·y2 + z2)2 – 4·y2·z2.

(2·y2)2 + 2·2·y2·z2 + (z2)2 - 2·2·y2 ·z2 = (2·y2 + z2)2 – 4·y2·z2.

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In the resulting expression, you can see the formula difference of squares:

(2·y2 + z2)2 – (2·y·z)2 = (2·y2 + z2 – 2·y·z)· (2·y2 + z2 + 2·y·z).

(2·y2 + z2)2 – (2·y·z)2 = (2·y2 + z2 – 2·y·z)· (2·y2 + z2 + 2·y·z).

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So, the method consists of two stages: allocation of the monomials of a complete square of m and n, the addition and subtraction of their double works. Selection method full square binomials can be used not only independently but also in combination with other methods: making the brackets common factor, change of variable, grouping terms, etc.

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Example 2.

Select the full square in the expression:

4·y2 + 2·y·z + z2.

Solution.

4·y2 + 2·y·z + z2 =[m = 2·y, n = z] = (2·y)2 + 2·2·y·z + (z) 2 – 2·y·z = (2·y + z)2 – 2·y·z.

Select the full square in the expression:

4·y2 + 2·y·z + z2.

Solution.

4·y2 + 2·y·z + z2 =[m = 2·y, n = z] = (2·y)2 + 2·2·y·z + (z) 2 – 2·y·z = (2·y + z)2 – 2·y·z.

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The method used for finding roots of a quadratic equation. The left part of the equation is a trinomial of the form a·y2 + b·y + c, where a, b and c are numbers and a ≠ 0.

a·y2 + b·y + c = a·(y2 + (b/a)·y) + c = a·(y2 + 2·(b/(2·a))·y) + c = a·(y2 + 2·(b/(2·a))·y + b2/(4·a2)) + c – b2/(4·a) = a·(y + b/(2·a)) 2 – (b2 – 4·a·c)/(4·a).

a·y2 + b·y + c = a·(y2 + (b/a)·y) + c = a·(y2 + 2·(b/(2·a))·y) + c = a·(y2 + 2·(b/(2·a))·y + b2/(4·a2)) + c – b2/(4·a) = a·(y + b/(2·a)) 2 – (b2 – 4·a·c)/(4·a).

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These calculations lead to the notion of discriminant which is equal to (b2 – 4·a·c)/(4·a) and the roots of the equation are equal:

y_1,2 = ±(b/(2•a)) ± √ ((b2 – 4·a·c)/(4·a)).

y_1,2 = ±(b/(2•a)) ± √ ((b2 – 4·a·c)/(4·a)).