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精英家教網 > 初中數學 > 題目詳情
閱讀并按要求完成各題.
(1)在化簡
7
-2
10
的過程中,小王和小李的化簡結果不一樣:
小王的化簡過程如下:
原式=
2-2
2×5
+5
=
(
2
)
2
-2
2
5
+(
5
)
2
=
(
2
-
5
)
2
=
2
-
5

小李的化簡過程如下:
原式=
(
2
)
2
-2
2
5
+(
5
)
2
=
(
2
-
5
)
2
=
5
-
2

請判斷誰的化簡結果正確,并說明理由.
(2)已知x=
6-2
5
,求(
1
x-2
+
1
x+2
)
x2-4
2(x-1)
的值.(結果保留根號)
分析:(1)根據二次根式的性質得到
(
2
-
5)2
=|
2
-
5
|=
5
-
2
,由此可判斷小李的化簡結果正確.
(2)理由(1)的方法可化簡x得到x=
5
-1,再對分式進行通分、約分得到原式=
x
x-1
,然后把x的值代入后解分母有理化即可.
解答:解:(1)小李的化簡結果正確.理由如下:
(
2
-
5)2
=|
2
-
5
|=
5
-
2

(2)x=
5-2
5
+1
=
(
5)2-2
5
+1
=
(
5
-1)2
=
5
-1,
原式=
x+2+x-2
(x-2)(x+2)
(x+2)(x-2)
2(x-1)

=
x
x-1

當x=
5
-1時,原式=
5
-1
5
-1-1
=3+
5
點評:本題考查了二次根式的化簡求值:先化簡再代入求值;二次根式運算的最后,注意結果要化到最簡二次根式,二次根式的乘除運算要與加減運算區分,避免互相干擾.也考查了分式的化簡求值.
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