【求完整說明及解答】中二數學題四條

1:
{4[(3y/2) + (1/6)]}²
= {4*(3y/2) + 4*(1/6)}² ...... 將 4 乘入括號中
= {6y + (2/3)}²
= (6y)² + 2(6y)(2/3) + (2/3)² ...... (a + b)² = a² + 2ab + b²
= 36y² + 8y + (4/9)
= (4/9)(81y² + 18y + 1)


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2:
(5x - 40y) [(x/7) + (8y/7)]
= 5x[(x/7) + (8y/7)] - 40y[(x/7) + (8y/7)] ...... 分配率：(5x - 40y)A = 5xA - 40yA
= [5x*(x/7) + 5x*(8y/7)] +[(-40y)*(x/7) + (-40y)*(8y/7)] ...... 分別將 5x 和 -40y 乘入括號中
= (5/7)x² + (40/7)xy - (40/7)xy - (320/7)y²
= (5/7)x² - (320/7)y²
= (5/7)(x² - 64y²)

另解：
(5x - 40y) [(x/7) + (8y/7)]
= 5(x - 8y) * (1/7)(x + 8y) ...... 分別抽出公因式 5 及 (1/7)
= (5/7) (x - 8y)(x + 8y)
= (5/7)(x² - 64y²) ...... (a - b)(a + b) = a² - b²


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3:
16 [(-2x/3) - (y/3)] [(7/4) - (x/2)]
= [16*(-2x/3) - 16*(y/3)] [(7/4) - (x/2)] ...... 將 16 乘入第一個括號中
= [(-32x/3) - (16y/3)] [(7/4) - (x/2)]
= (-32x/3)[(7/4) - (x/2)] - (16y/3)] [(7/4) - (x/2)] ...... 分配率：(a - b)(c - d) = a(c - d) -b(c - d)
= (-32x/3)*(7/4) - (-32x/3)*(x/2) - (16y/3)*(7/4) + (16y/3)*(x/2) ..... 分別將 (-32x/3) 和 -(16y/3) 乘入括號中
= -(56/3)x + (16/3)x² - (28/3)y + (8/3)xy
= (4/3)(-14x + 4x² - 7y + 2xy)

另解：
16 [(-2x/3) - (y/3)] [(7/4) - (x/2)]
= 16 * (-1/3)[2x + y] * (1/4)[7 - 2x] ...... 分別抽出 (-1/3) 和 (1/4)
= -(4/3)(2x + y)(7 - 2x)
= -(4/3)[2x(7 - 2x) + y(7 - 2x)] ...... 分配率：(a + b)(c - d) = a(c - d) +b(c - d)
= -(4/3)(14x - 4x² + 7y - 2xy) ...... 將 2x 和 y 乘入括號中
= (4/3)(-14x + 4x² - 7y + 2xy)


4:
(x + 2y - 3z)(x + 2y + 3z)
= x(x + 2y + 3z) + 2y(x + 2y + 3z) - 3z(x + 2y + 3z) ...... 分配率： (x + 2y - 3z)A = xA + 2yA- 3zA
= x² + 2xy + 3xz + 2xy + 4y² + 6yz - 3xz - 6yz - 9z² ...... 分別將 x、2y 和 -3z 乘入括號中
= x² + 4y² - 9z² + 4xy

1.
因為(x+y)^2 = x^2+y^2+2xy

所以[4(3y/2 + 1/6)]^2 先拆細括號
= (6y+2/3)^2
= 36y^2+8y+4/9

2. (5x - 40y)(x/7 + 8y/7)
=5x(x/7 + 8y/7)+(-40y)(x/7 + 8y/7) 先拆括號
=5/7x^2+40/7xy-40/7xy-320/7y^2 留意正負號雙乘之後的結果
=5/7x^2-320/7y^2
=5/7(x^2-64y^2) ,因為(a+b)(a-b)=(a^2-b^2)
=5/7(x+8y)(x-8y)

3. 16(-2x/3 - y/3)(7/4 - x/2)
(-32/3x-16/3y)(7/4 - x/2) 做法跟第2題類似
(-32/3x-16/3y)(7/4)+(-32/3x-16/3y)(- x/2)
-56/3x-28/3y+16/3x^2+8/3xy

4.因為(a+b)(a-b)=(a^2-b^2)
所以a=x+2y
b=3z
(x + 2y - 3z)(x + 2y + 3z)
(x+2y)^2-(3z)^2
= x^2+4xy+4y^2-9z^2

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