數學49條....

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1) (x+y)^2+(x-y)^2
= (x²+2xy+y²)+(x²-2xy+y²)
= 2x²+2y²
= 2(x²+y²)

2) (2x+3y)^2+(2x-3y)^2
= (4x²+12xy+9y²)+(4x²-12xy+9y²)
= 8x²+18y²
= 2(4x²+9y²)

3) (x+y)^2-(x-y)^2
= (x²+2xy+y²)-(x²-2xy+y²)
= 2xy-(-2xy)
= 4xy

4) (2x+3y)^2-(2x-3y)^2
= (4x²+12xy+9y²)-(4x²-12xy+9y²)
= 12xy-(-12xy)
= 24xy

5) (x+5)^2-(x-5)^2
= (x²+10x+25)-(x²-10x+25)
= 10x-(-10x)
= 20x

6) (ax+by)^2-(ax-by)^2
= (a²x²+2abxy+b²y²)-(a²x²-2abxy+b²y²)
= 2abxy-(-2abxy)
= 4abxy

7) [2(x+y)]^2
= 2² (x+y)²
= 4(x²+2xy+y²)

8) [1/2(x+y)]^2
= (1/2)² (x+y)²
= 1/4 (x²+2xy+y²)
= 1/4 x² + 1/2 xy + 1/4 y²

9) [1/3(2a-b)]^2
= (1/3)² (2a-b)²
= 1/9 (4a²-4ab+b²)
= 4/9 a² - 4/9 ab + 1/9 b²

10) [-2(x+y)]^2
= (-2)² (x+y)²
= 4 (x²+2xy+y²)

11) [-1/6(2a-3b)]^2
= (-1/6)² (2a-3b)²
= 1/36 (4a²-12ab+9b²)
= 1/9 a² - 1/3 ab + 1/4 b²

12) (x+y)(x-y)
= x²+xy-xy-y²
= x² - y²

13) (x+4)(x-4)
= x² - 16 (refer to #12)

14) (2x+5)(2x-5)
= 4x² - 25 (#12)

15) (x+2y)(x-2y)
= x² - 4y² (#12)

16) (2x+3y)(2x-3y)
= 4x² - 9y² (#12)

17) (5x-6y)(5x+6y)
= 25x² - 36y² (#12)

18) (xy-1)(xy+1)
= x²y² - 1

19) (x+2/3y)(x-2/3y)
= x² - (2/3 y)²
= x² - 4/9 y²

20) (-x-2)(-x+2)
= (-x)² - 2²
= x² - 4

21) (-4a+6)(-4a-6)
= (-4a)² - 6²
= 16a² - 36
= 4(4a² - 9)

22) (-3x+ab)(-3x-ab)
= (-3x)² - (ab)²
= 9x² - a²b²

23) 2(4x-3)(4x+3)
= 2[(4x)² - 3²]
= 2(16x² - 9)
= 32x² - 18

24) -3(a+3x)(a-3x)
= -3[a² - (3x)²]
= -3(a²-9x²) = 3(9x²-a²)
= 27x² - 3a²

25) -5(5x+1)(5x-1)
= -5[(5x)² - 1²]
= -5(25x² - 1) = 5 (1-25x²)
= 5 - 125x²

26) -2(-a-4)(-a+4)
= -2[(-a)²-4²]
= -2(a²-16) = 2(16-a²)
= 32 - 2a²

27) 2(2/3x-3/4)(2/3x+3/4)
= 2[(2/3x)²-(3/4)²]
= 2(4/9 x² - 9/16)
= 8/9 x² - 9/8

28) (x+5)^2+(x+3)(x-3)
= (x²+10x+25)+(x²-3²)
= 2x² + 10x + 34
= 2(x²+5x+17)

29) (3x-y)^2-(x+y)(x-y)
= (9x²-6xy+y²)-(x²-y²)
= 8x²-6xy

30) (x-2)^2-4(2x+5)(2X-5)
= (x²-4x+4)-4[(2x)²-5²)]
= (x²-4x+4)-4(4x²-25)
= (x²-4x+4)-(16x²-100)
= -15x² - 4x + 104

31) (x+3)(x+5)
= x² + (3+5)x + 15
= x² + 8x + 15

32) (x+7)(x-2)
= x² + (7-2)x - 14
= x² + 5x - 14

33) (x+2)(x-5)
= x² + (2-5)x - 10
= x² - 3x - 10

34) (x-3)(x-7)
= x² + (-3-7)x + 21
= x² - 10x + 21

35) (x+3)(x+8)
= x² + 11x + 24

36) (x+6)(x-2)
= x² + 4x - 12

37) (x-5)(x+1)
= x² - 4x - 5

38) (x-3)(x-5)
= x² - 8x + 15

39) (x+3y)(x+y)
= x²+ 4xy + 3y²

40) (x+3y)(x-2y)
= x²+ xy - 6y²

41) (x-4y)(x+6y)
= x² + 2xy - 24y²

42) 2(x+2)(x-3)
= 2(x² - xy - 6)
= 2x² - 2xy - 12

43) -3(x-3)(x+4)
= -3(x² + x - 12)
= -3x² - 3x + 36

44) -(xy-4)(xy-7)
= -[(xy)² - 11xy + 28]
= -x²y² + 11xy - 28

45) 2(x-5y)(x+10y)
= 2[x+5xy-50y²]
= 2x + 10xy - 100y²

46) (x+3)(x-7)+(x+7)(x-3)
= (x²-4x-21)+(x²+4x-21)
= 2x²-42
= 2(x²-21)

47) (x-3)(x+5)-(x+1)^2
= (x²+2x-15)-(x²+2x+1)
= -16

48) (x+3)(x-5)-(x+4)(x-4)
= (x²-5x-15)-(x²-16)
= -5x-15-(-16)
= 1 - 5x

49) (x+6)(x-6)-(x+9)(x-4)
= (x²-36)-(x²+5x-36)
= -5x