The addition of these two formulas is called the complete square formula. The square of the sum (or difference) of two numbers is equal to the sum of the squares of these two numbers, plus (or minus) twice the product of these two numbers.
We usually express it as:
(a b) 2 = a 22ab+b 2
note:
Generally, A and B are algebraic expressions representing a whole, not necessarily numbers, such as [(3x-y)-(2x+2y)] [(3x-y)+(2x+2y)] = 5x2+6xy+y2.
The common mistakes in the complete square formula are: ① it is difficult for students to jump out of the original fixed thinking, such as typical mistakes; (Error reason: analogy based on the formula, and "creation" at will) ② Confuse the formula; (3) The symbol in the operation result is wrong; ④ Variant application is difficult to master.
First, understand the characteristics of the left and right sides of the formula (1), learn to deduce the formula (these two formulas are obtained according to the meaning of power and the multiplication rule of polynomial), and truly understand the inaccuracy of random "creation"; (2) Learn to summarize the meaning of the formula in words: the square of the sum (or difference) of two numbers is equal to the sum of their squares, plus (or minus) twice their product. Sum and sum are both called complete square formulas. In order to distinguish them, we call the former the complete square formula of the sum of two numbers and the latter the complete square formula of the difference of two numbers. (3) The structural features of these two formulas are: 1, 0. 2. When the two symbols on the left are the same, all the symbols on the right are connected with "+"; When the two symbols on the left are opposite, the square items on the right are connected by "+"and then multiplied by "-"(note: the symbol of this item is not included here); 3. The letters in the formula can represent a specific number (positive or negative), and can also represent mathematical formulas such as monomial or polynomial. (4) Unity of the two formulas: because the two formulas can actually be regarded as one formula: the complete square formula of the sum of two numbers. This can not only prevent the confusion of formulas, but also prevent the error of operation symbols. Second, grasp the four steps of using the formula: 1, "test": when calculating, we must first observe whether the characteristics of the topic meet the conditions of the formula. If not, first convert it into a form that meets the conditions of the formula, and then use the formula to calculate. If not, we should use the corresponding multiplication rule to calculate. Second, "derivation": The key to correctly choose the complete square formula is to determine the formula. 4. "Check": Learn to check after completing the operation, not only to go back and reflect on whether the calculation basis and symbols of each step are correct, but also to check with the multiplication rule of polynomials to ensure foolproof. Third, master the four changes of the formula (1) and change the sign: Example 1: Calculate by using the complete square formula: (1) (2) Analysis: This example changes the signs of A and B in the formula. One of the processing methods: converting the two formulas into a multiplexing formula (reflection); Method 2: Convert the two formulas into: directly calculate with the formula; Method 3: Convert the two formulas into: directly use the formula to calculate (this method is based on the unification of the two formulas, which is easy to understand and will not be confused); (2) Number of variables: Example 2: Calculation: Analysis: The left side of the complete square formula is the multiplication of two identical binomials, but there are three items in this example, so we should consider merging two of them with the overall idea into one, so as to resolve the contradiction. Therefore, when using the formula, it can be converted into or or first, and then calculated. (3) Variant structure example 3: Calculate by formula: (1) (x+y) (2x+2y); (2)(a+b)(-a-b)； (3) (a-b) (b-a) analysis; In this example, all binomials are multiplied by binomials. On the surface, the appearance structure does not conform to the characteristics of the formula, but after careful observation, it is easy to find that one of the factors can be appropriately deformed, namely (1) (x+y) (2x+2y) = 2 (x+y)? ; (2)(a+b) (-a-b)=-(a+b)？ ; (3)(a-b) (b-a)=-(a-b)？ (4) Simple operation Example 4: Calculation: (1) 9992 (2) 100. 1 2 Analysis: In this example, 999 is close to 1000, 100. Namely: (1). Fourth, learn formula 1 expansion and formula mixing. Example 5: Calculation: (l) (x+y+z) (x+y-z) (2) (2x-y+3z) (y-3z-2x) Analysis: This example is a trinomial multiplied by a trinomial, which has the following characteristics: Therefore, we can consider combining items with the same term and the opposite number into a square difference. Namely: (1) (x+y+z) (x+y-z) = [(x+y)+z] [(x+y)-z] = … (2) (2x-y+3z) (y-3z+2x). Find the following values: (1) a2+b2; (2) (A-B) 2 Analysis: This case is a typical algebraic expression evaluation problem. According to conventional thinking, it is difficult to find the values of a and b respectively; Through careful exploration, it is easy to combine these conditions with the complete square formula, and it is easy to find a solution to the problem by using the variant of the complete square formula. Namely: (1) a2+B2 = (a+b) 2-2ab = … (2) (a-b) 2 = (a+b) 2-4ab = … 3. Inverse use of formula: Example 7: Calculation: Analysis: It is more complicated to use multiplication formula and rules directly in this question.
(a+b)(a-b)=a^2-b^2
The product of the sum of two numbers and the difference between the two numbers is equal to the square difference between the two numbers. This formula is called the square difference formula of multiplication.
When multiplication is the sum of two numbers and the difference between the two numbers is multiplied, the product is binomial. This is because when two binomials with such characteristics are multiplied, two of the four terms of the product will be opposite. The result of the merger of these two items is zero, so there are only two items left. And their product is equal to the square difference of these two numbers in the multiplication formula, that is, A-B = (A+B) (A-B).
The sum of the differences between two numbers is equal to their square difference.
[Inverse derivation of square difference formula]
a^2-b^2
=a^2-b^2+(ab-ab)
=(a^2-ab)+(ab-b^2)
=a(a-b)+b(a-b)
=(a+b)(a-b)
Formula application
[solving equation]
x×x-y×y= 199 1
[train of thought analysis]
Solve by square difference formula
[Problem solving process]
x^2-y^2= 199 1
(x+y)(x-y)= 199 1
Because 199 1 can be divided into 1× 199 1,/kloc-0 /×1.
So if x+y = 199 1 and x-y = 1, the solution is x = 996 and y = 995.
If x+y = 18 1, x-y =1,x = 96, y = 85 can also be negative.
So the solution is x = 996, y = 995, or x = 996, y =-995, or x =-996, y = 995 or x =-996, y =-995.
Or x = 96, y = 85, or x = 96, y =-85 or x =-96, y = 85 or x =-96, y =-85.