题目内容
计算和化简:(1)计算:
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(2)已知x2-3=5x,求(x-1)(2x-1)-(x+1)2+1的值;
(3)先将
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【答案】分析:(1)根据二次根式的化简根式
=|a|,以及
=
•
(a≥0,b≥0)把原式的第一项化为最简二次根式,第二项利用特殊角的三角函数值化简,第三项根据a=1(a≠0)进行化简,最后一项利用a-p=
(a≠0)进行化简,然后把同类二次根式合并即可得到结果;
(2)把所求式子利用多项式的乘法法则及完全平方公式化简,合并同类项后,由已知的等式得出x2-5x的值,代入化简的式子中即可求出值;
(3)把第一个因式的分子分母分别分解因式,第二个因式通分后,利用同分母分式的减法法则:分母不变只把分子相减进行计算,两因式约分可得出最简结果,选择一个合适的x的值代入化简后的式子中即可求出值.
解答:解:(1)
=2
-2×
+1-3
=2
-
-2
=
-2;
(2)∵x2-3=5x,
∴x2-5x=3,
则(x-1)(2x-1)-(x+1)2+1
=2x2-x-2x+1-(x2+2x+1)+1
=2x2-x-2x+1-x2-2x-1+1
=x2-5x+1
=3+1
=4;
(3)
=
•
=x+2,
当x=2时,原式=x+2=2+2=4.
点评:此题考查了分式的化简求值,实数的运算,以及整式的混合运算,在进行分式化简求值时,加减的关键是通分,通分的关键是找各分母的最简公分母;分式的乘除运算关键是约分,约分的关键是找公因式,若分子分母中出现多项式,应将多项式分解因式后再约分.其中第二小题利用了整体代入的思想,第三小题在选择x的值时应注意分式的分母不能为0,即x≠0,且x≠1.
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(2)把所求式子利用多项式的乘法法则及完全平方公式化简,合并同类项后,由已知的等式得出x2-5x的值,代入化简的式子中即可求出值;
(3)把第一个因式的分子分母分别分解因式,第二个因式通分后,利用同分母分式的减法法则:分母不变只把分子相减进行计算,两因式约分可得出最简结果,选择一个合适的x的值代入化简后的式子中即可求出值.
解答:解:(1)

=2
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=2
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=
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(2)∵x2-3=5x,
∴x2-5x=3,
则(x-1)(2x-1)-(x+1)2+1
=2x2-x-2x+1-(x2+2x+1)+1
=2x2-x-2x+1-x2-2x-1+1
=x2-5x+1
=3+1
=4;
(3)
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=
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=x+2,
当x=2时,原式=x+2=2+2=4.
点评:此题考查了分式的化简求值,实数的运算,以及整式的混合运算,在进行分式化简求值时,加减的关键是通分,通分的关键是找各分母的最简公分母;分式的乘除运算关键是约分,约分的关键是找公因式,若分子分母中出现多项式,应将多项式分解因式后再约分.其中第二小题利用了整体代入的思想,第三小题在选择x的值时应注意分式的分母不能为0,即x≠0,且x≠1.
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