two form 2 question about identities

知識就是力量

2007-11-30 8:10 pm

Show all the necessary steps.
1. Prove that the following equation is identity
(x+3y)/2 - (2x-y)3 + (x-5y)/6 = y
2(a). Prove that (m+1)2 – 3m – 1 = m2 – m is an identity.
(b). By substituting m = x - 1 into the identity in (a), find the unknown constants A and B in x2 – 3x + 2 = (Ax – B)(x – 2) ← the equal sign is 3 lines that one(is identity to)

回答 (1)

Teddy

2007-11-30 8:47 pm

✔ 最佳答案

1)
(x+3y)/2 - (2x-y)/3 + (x-5y)/6 = y
左手邊...
= (x+3y)/2 - (2x-y)/3 + (x-5y)/6
= 3(x+3y)/6 - 2(2x-y)/6 + (x-5y)/6
= [ 3(x+3y) - 2(2x-y) + (x-5y) ]/6
= (3x + 9y - 4x + 2y + x - 5y)/6
= 6y/6
= y
= 右手邊
所以, 數式兩邊相等

2a)
(m+1)^2 - 3m - 1 = m^2 - m
左手邊...
= (m+1)^2 - 3m - 1
= m^2 + 2m + 1 - 3m - 1
= m^2 - m
= 右手邊
所以, 數式兩邊相等

b)
代入 m = x-1
數式左手邊會變成...
(x-1+1)^2 - 3(x-1) - 1
= x^2 - 3x + 3 - 1
= x^2 - 3x + 2
= (x-2)(x-1)

題目所說...("=" 代表全等)
x^2 - 3x + 2 "=" (Ax-B)(x-2)
(x-2)(x-1) "=" (Ax-B)(x-2)
所以...
Ax - B "=" x - 1
A = 1
B = 1

參考: Me

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