配方法是数学中重要的一种思想方法.它是指将一个式子的某一部分通过恒等变形化为完全平方式或几个完全平方式的和的方法.这种方法常被用到代数式的变形中,并结合非负数的意义来解决一些问题.我们定义:一个整数能表示成
(a、b是整数)的形式,则称这个数为“完美数”.例如,5是“完美数”.理由:因为
,所以5是“完美数”.
解决问题:
(1)已知10是“完美数”,请将它写成
(a、b是整数)的形式__________;
(2)若
可配方成
(m、n为常数),则
________;
探究问题:
(3)已知
,则
____________;
(4)已知
(x、y是整数,k是常数),要使S为“完美数”,试求出符合条件的一个k值,并说明理由.
拓展结论:
(5)已知实数x、y满足
,求
的最值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c925be255ca736a53b24d13ddede1a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa0f7368432b977563895c2d28862766.png)
解决问题:
(1)已知10是“完美数”,请将它写成
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c925be255ca736a53b24d13ddede1a86.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00e592d76fd3ef30e964393a1eff2624.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/855c1b3733e046c292c4e166954ef216.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/549f99e6e10e61af2e7734c4d01ea90c.png)
探究问题:
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e1fbaeb255be226a7562f2e5762a99b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edff1881635893293dd411ead8194aca.png)
(4)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df8141fb6d6b201d14cd242fb11ba308.png)
拓展结论:
(5)已知实数x、y满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ca3db78d6e9eded9416c8dd6f34577.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f0b97fdfebd0cd0d78e78df1e77c5b.png)
22-23九年级上·广东梅州·期中 查看更多[4]
广东省梅州市五华县2022-2023学年九年级上学期期中考试数学试题安徽省天长市实验中学2022-2023学年八年级下学期第一阶段质量检测数学试卷浙江省金华市东阳市江北初级中学等四校联2022-2023学年八年级下学期期中数学试题(已下线)专题05用配方法求解一元二次方程(3个知识点7种题型2个易错点4种中考考法)-【帮课堂】2023-2024学年九年级数学上册同步学与练(北师大版)
更新时间:2022-11-14 09:48:52
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【推荐1】[知识重现]
我们知道,在
中,
已知底数
指数
,求幂
的运算叫做乘方运算,例如
;
已知幂
,指数
,求底数
的运算叫做开方运算,例如![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3604c089eceb6f7c670fddb1550997b.png)
[学习新知]
现定义:如果
且
,即
的
次方等于
且
),那么数
叫做以
为底
的对数
,记作
.其中
叫做底数,
叫做真数,
叫做以
为底
的对数.例如:若
,则
(其中
为底数,
为真数)注意:零没有对数;在实数范围内,负数没有对数.
[应用新知]
(1)填空:
在
中,已知幂
,底数
且
),求指数
的运算叫做____运算;
(2)选择题:在式子
中,真数是( )
A.
B.
C.
D.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9740bae87cbcfb6d389145d64cfd42ba.png)
(3)①求出下列各对数的值:
;
;
.
②根据①中计算结果,请直接写出
之间的关系.(其中
且
)
我们知道,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90b5a213235f7d61e6b1c4663ad8d773.png)
已知底数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe9fbbd9c88736e500f5251f97b08452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feae878a430a1b23774127547d842cb6.png)
已知幂
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3604c089eceb6f7c670fddb1550997b.png)
[学习新知]
现定义:如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d15fb41199ee58d61b3fc18da15d7bf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37fa1476cf3552b9ae91ef039b1c6c80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b0254319bdec95aa8db0932d1a29cb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1481d5c423b5f4f9fb960a7c63d2dc1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65130628c4328a2144486245b7a14a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feae878a430a1b23774127547d842cb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7950d4655c518c4067c345babb0de73d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3304e23f3b0f9569c4140ca89b6498.png)
[应用新知]
(1)填空:
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90b5a213235f7d61e6b1c4663ad8d773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efbcf4f4a9d7c6f3114547997ea2388d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)选择题:在式子
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf92cb757285d894da1d58b35fe03400.png)
A.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9740bae87cbcfb6d389145d64cfd42ba.png)
(3)①求出下列各对数的值:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9589d5c348d2f40f6ae05d56840b13e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c9253bae4c87a03b512718b2f04a097.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aedf373b8747d8815c98519aa015b44.png)
②根据①中计算结果,请直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3430fcf3144e6f626494530184e0c412.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b676b72370885a1b4c5a182d006cace.png)
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【推荐2】定义:对任意一个三位数a,如果a满足百位数字与十位数字相同,个位数字与十位数字不相同,且都不为零,那么称这个三位数为“半异数”,将一个“半异数”的各个数位上的数字交换后得到新的三位数,把所有的新三位数的和与111的商记为f(a).例如:a=112,a为“半异数”,将a各个数位上的数字交换后得到新的三位数有121、211、112,所有新三位数的和为121+211+112=444,和与111的商为444÷111=4.所以f(112)=4,根据以上定义,回答下列问题:
(1)计算f(227);
(2)数p,q是两个三位数,它们都有“半异数”,P的个位数字是3,q的个位数字是5,p≤q.规定,k=
,若f (p)+f(q)的和是13的倍数,求k的最大值.
(1)计算f(227);
(2)数p,q是两个三位数,它们都有“半异数”,P的个位数字是3,q的个位数字是5,p≤q.规定,k=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a507c062709cfe2f218896247461c7d3.png)
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【推荐1】阅读下面的解答过程,求y2+4y+8的最小值.解:y2+4y+8=y2+4y+4+4=(y+2)2+4,∵(y+2)2≥0,∴(y+2)2+4≥4,∴y2+4y+8的最小值为4.仿照上面的解答过程,求x2-x+4的最小值和6-2x-x2的最大值.
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【推荐2】阅读材料:
把代数式通过配凑等手段得到局部完全平方式,再进行有关计算和解题,这种解题方法叫做配方法
如(1)用配方法分解因式:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1efa501fd68ec17b7c63019ad85604a0.png)
解:原式=![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95ad0b43da3c6d9153e4e8f1af9b648c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51673ba49a5cbdd94347d633c2750815.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0512c301ae803abab6235b55ec794e9.png)
=![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b92f2aea45b4226ea47171c2d98c61b.png)
(2)M=
,利用配方法求M的最小值
解:M=![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29351b1b077b2a52cd604606115017ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1478ecf995cfa038e7e3a4d484df99c2.png)
=![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4aefc6636261c394d3e20f92200a98c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/144c5651cb21733c184e10188c84ec6c.png)
当
时,M有最小值1
请根据上述材料,解决下列问题:
(1)在横线上添加一个常数,使之成为完全平方式:
(2)用配方法分解因式:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0094f10070ba22d29f65d257338df224.png)
(3)若M=
,求M的最小值.
把代数式通过配凑等手段得到局部完全平方式,再进行有关计算和解题,这种解题方法叫做配方法
如(1)用配方法分解因式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1efa501fd68ec17b7c63019ad85604a0.png)
解:原式=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95ad0b43da3c6d9153e4e8f1af9b648c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51673ba49a5cbdd94347d633c2750815.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0512c301ae803abab6235b55ec794e9.png)
=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b92f2aea45b4226ea47171c2d98c61b.png)
(2)M=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29351b1b077b2a52cd604606115017ac.png)
解:M=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29351b1b077b2a52cd604606115017ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1478ecf995cfa038e7e3a4d484df99c2.png)
=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4aefc6636261c394d3e20f92200a98c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/144c5651cb21733c184e10188c84ec6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c1b60761984cd5298f738772b7af2e0.png)
请根据上述材料,解决下列问题:
(1)在横线上添加一个常数,使之成为完全平方式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d9b3622672006527a5803a5ee7005b9.png)
(2)用配方法分解因式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0094f10070ba22d29f65d257338df224.png)
(3)若M=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c47486d04ed7e9c80da419e99581d54.png)
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