What are the complete square formulas in mathematics?

The common variations of 1. complete square formula are as follows

a2+b2=(a+b)2-2ab，

a2+b2=(a-b)2+2ab，

(a+b)2-(a-b)2=4ab，

a2+b2+c2=(a+b+c)2-2(ab+ac+bc)

Second, the application of multiplication formula deformation

Example 1: It is known that x2+y2+4x-6y+ 13=0, and both x and y are rational numbers. Find the value of xy.

Analysis: Reverse the complete power formula and put it

X2+y2+4x-6y+ 13 is the sum of two completely flat modules, and the values of x and y can be obtained by using the nonnegativity of completely flat modules.

Solution: ∫x2+y2+4x-6y+ 13 = 0,

(x2+4x+4)+(y2-6y+9)=0，

That is, (x+2)2+(y-3)2=0.

∴x+2=0,y=3=0。

That is, x=-2 and y=3.

∴xy=(-2)3=-8。

Analysis: This question is skillfully used.

Example 3 is known: a+b=8, ab= 16+c2. Find the value of (a-b+c)2002.

Analysis: The value of (a-b+c)2002 cannot be directly derived from the known conditions. We can use (a-b)2=(a+b)2-4ab to determine the relationship between A-b and C, and then calculate the value of (a-b+c) 2002.

Solution: (a-b) 2 = (a+b) 2-4ab = 82-4 (16+C2) =-4c2.

Namely: (a-b)2+4c2=0.

∴a-b=0,c=0。

∴(a-b+c)2002=0。

Example 4 shows that A, B, C and D are positive rational numbers and satisfy a4+b4+C4+D4=4abcd.

Proof: a = b = c = d.

Analysis: From the characteristics of a4+b4+C4+D4=4abcd, we can see that it can be transformed into a complete square, and then find a proof idea.

Proof: ∫a4+B4+C4+D4 = 4 ABCD

∴a4-2a2b2+b4+c4-2c2d2+d4+2a2b2-4abcd+2c2d2=0，

(a2-b2)2+(c2-d2)2+2(ab-cd)2=0 .

a2-b2=0，c2-d2=0，ab-cd=0

And ∵a, b, c and d are positive rational numbers,

∴a=b,c=d。 Substitute ab-cd=0,

A2=c2, that is, A = C. ..

So there is a = b = c = d.

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