一元二次方程知识点总结+练习

一元二次方程知识点总结练习一元二次方程的解法【知识点归纳与总结】2一、概念:一元二次方程的一般形式为:ax+bx+c=0,(a?0)，它是只含一个未知数，并且未知数的最高次数是2的整式方程。二、基本思路与方法:解一元二次方程的基本思想方法是通过“降次”将它化为两个一元一次方程。一元二次方程有四种解法:1、直接开平方法;2、配方法;3、公式法;4、因式分解法。21用直接开平方法解形如(x-m)=n(n?0)的方程，其解为x=m?.722例1(解方程(1)(3x+1)=7(2)9x-24x+16=11522(配方法:用配方法解方程ax+bx+c=0(a?0)2先将常数c移到方程右边:ax+bx=-cbc2将二次项系数化为1:x+x=-aabbcb222方程两边分别加上一次项系数的一半的平方:x+x+()=-+()a2aa2a22方程左边成为一个完全平方式:(x+)=当b-4ac?0时，x+=??x=(这就是求根公式)2例2(用配方法解方程3x-4x-2=0223(公式法:把一元二次方程化成一般形式，然后计算判别式?=b-4ac的值，当b-4ac?0时，把各项系数a,b,c2的值代入求根公式x=(b-4ac?0)就可得到方程的根。2例3(用公式法解方程2x-8x=-5toliftconstruction.3,constructionofroadpavementarenecessary,cleanandfreeofwater,drainagefacilitiesingoodcondition,smoothdrainage.4,FengShuipipeline,communicationfacilities,constructionlightinglayoutreasonable,markclear.5,pointparkconstructionmachineryandequipment,materials,toolsplacedinanorderlymanner,workisfinished,andvehicleappearancelooksclean,fire-fightingequipmentavailable,unobstructed.6,theconstructionofvarioustypesofscaffolding,channel,ladder,guardrails,safetynetsandothersecurityprotectionfacilitiestoimprovereliability,securityIDvisible.7,activelycarryoutdust,toxicity,noisecontrolandarrangetogiveupballastwater.8,livingquartersclean,beautifulenvironment.Constructionsafetyplan15.215.2.1safetyguidelinesthesafetyapproach:people-oriented,fullparticipation,buildasafetyculture,anddynamic.15.2.2safetygoalourcompanyattachesspecialimportancetothisproject,thepromotionofcomprehensivesecurityinthemanagement,constructionmanagement,realization:nopersonaldeathbyaccident;Nomajorconstructionmachineryandequipmentdamageaccidents;Nomajorfireaccidents;Notprimarilyresponsiblefortheoccurrenceofmajoraccidents;Nomajorcollapseandrooffall,oftheaccident;Noenvironmentalpollutionaccidentandoccupationalhealthinjuries,foodpoisoningincidents.Controltheseriousinjuryaccidentrateislessthan4(因式分解法:把方程变形为一边是零，把另一边的二次三项式分解成两个一次因式的积的形式，让两个一次因式分别等于零，得到两个一元一次方程，解这两个一元一次方程所得到的根，就是原方程的两个根。这种解一元二次方程的方法叫做因式分解法。例4(用因式分解法解下列方程:2(1)(x+3)(x-6)=-8(2)2x+3x=022(3)6x+5x-50=0(4)x-2(+)x+4=0小结:一般解一元二次方程，最常用的方法还是因式分解法，在应用因式分解法时，一般要先将方程写成一般形式，同时应使二次项系数化为正数。直接开平方法是最基本的方法。公式法和配方法是最重要的方法。公式法适用于任何一元二次方程(有人称之为万能法)，在使用公式法时，一定要把原方程化成一般形式，以便确定系数，而且在用公式前应先计算判别式的值，以便判断方程是否有解。配方法是推导公式的工具，掌握公式法后就可以直接用公式法解一元二次方程了，所以一般不用配方法解一元二次方程。但是，配方法在学习其他数学知识时有广泛的应用，是初中要求掌握的三种重要的数学方法之一，一定要掌握好。(三种重要的数学方法:换元法，配方法，待定系数法)。例5(用适当的方法解下列方程。222(1)4(x+2)-9(x-3)=0(2)x-5x+6=022(3)2x-3x=4(4)4x-x+6=0toliftconstruction.3,constructionofroadpavementarenecessary,cleanandfreeofwater,drainagefacilitiesingoodcondition,smoothdrainage.4,FengShuipipeline,communicationfacilities,constructionlightinglayoutreasonable,markclear.5,pointparkconstructionmachineryandequipment,materials,toolsplacedinanorderlymanner,workisfinished,andvehicleappearancelooksclean,fire-fightingequipmentavailable,unobstructed.6,theconstructionofvarioustypesofscaffolding,channel,ladder,guardrails,safetynetsandothersecurityprotectionfacilitiestoimprovereliability,securityIDvisible.7,activelycarryoutdust,toxicity,noisecontrolandarrangetogiveupballastwater.8,livingquartersclean,beautifulenvironment.Constructionsafetyplan15.215.2.1safetyguidelinesthesafetyapproach:people-oriented,fullparticipation,buildasafetyculture,anddynamic.15.2.2safetygoalourcompanyattachesspecialimportancetothisproject,thepromotionofcomprehensivesecurityinthemanagement,constructionmanagement,realization:nopersonaldeathbyaccident;Nomajorconstructionmachineryandequipmentdamageaccidents;Nomajorfireaccidents;Notprimarilyresponsiblefortheoccurrenceofmajoraccidents;Nomajorcollapseandrooffall,oftheaccident;Noenvironmentalpollutionaccidentandoccupationalhealthinjuries,foodpoisoningincidents.Controltheseriousinjuryaccidentrateislessthan分析:(1)首先应观察题目有无特点，不要盲目地先做乘法运算。观察后发现，方程左边可用平方差公式分解因式，化成两个一次因式的乘积。(2)可用十字相乘法将方程左边因式分解。(3)化成一般形式后利用公式法解。(4)利用公式法解。一元二次方程根的判别式一、知识要点:221.一元二次方程ax+bx+c=0(a?0)的根的判别式Δ=b-4ac。Δ>0时，方程有两个不相等的实数根。Δ=0时，方程有两个相等的实数根。Δ<0时，方程没有实数根。2222判断2x+x+4=0，4x+3x-5=0，x+5x+2=0，2x+x+3=5有没有实数根。以上定理也可以逆向应用。在应用判别式之前，要把方程化为一般形式，以便正确找出a、b、c的值。2注意:(1)根的判别式是指Δ=b-4ac，不是Δ=，(2)使用判别式之前一定要先把方程变为一元二次方程的一般形式。2.根的判别式有以下应用:?不解一元二次方程，判断根的情况。?根据方程根的情况，确定待定系数的取值范围。?证明字母系数方程有实数根或无实数根。2注意:?如果说方程有实数根，即应当包括有两个不等实根或有两相等实根两种情况，此时b-4ac?0，切勿丢掉等号。2?根的判别式b-4ac的使用条件，是在一元二次方程中，而非别的方程中，因此，要注意隐含条件a?0.二、例题精讲:例1(不解方程，判断下列方程的根的情况:22222(1)2x+3x-4=0(2)3x+2=2x(3)x+1=x(4)ax+bx=0(a?0)(5)ax+c=0(a?0)toliftconstruction.3,constructionofroadpavementarenecessary,cleanandfreeofwater,drainagefacilitiesingoodcond3ition,smoothdrainage.4,FengShuipipeline,communicationfacilities,constructionlightinglayoutreasonable,markclear.5,pointparkconstructionmachineryandequipment,materials,toolsplacedinanorderlymanner,workisfinished,andvehicleappearancelooksclean,fire-fightingequipmentavailable,unobstructed.6,theconstructionofvarioustypesofscaffolding,channel,ladder,guardrails,safetynetsandothersecurityprotectionfacilitiestoimprovereliability,securityIDvisible.7,activelycarryoutdust,toxicity,noisecontrolandarrangetogiveupballastwater.8,livingquartersclean,beautifulenvironment.Constructionsafetyplan15.215.2.1safetyguidelinesthesafetyapproach:people-oriented,fullparticipation,buildasafetyculture,anddynamic.15.2.2safetygoalourcompanyattachesspecialimportancetothisproject,thepromotionofcomprehensivesecurityinthemanagement,constructionmanagement,realization:nopersonaldeathbyaccident;Nomajorconstructionmachineryandequipmentdamageaccidents;Nomajorfireaccidents;Notprimarilyresponsiblefortheoccurrenceofmajoraccidents;Nomajorcollapseandrooffall,oftheaccident;Noenvironmentalpollutionaccidentandoccupationalhealthinjuries,foodpoisoningincidents.Controltheseriousinjuryaccidentrateislessthan222例2(求证方程(m+1)x-2mx+(m+4)=0没有实数根。2例3(已知关于x的方程kx-4kx+k-5=0有两个相等的实数根，求k的值并解这个方程。一元二次方程的应用一、握手问题1、某小组同学元旦互赠贺年卡一张，全组共赠贺年卡90张，这个小组有几位同学,2、某校开展足球比赛，每个班组织一个班队，在第一轮比赛中，每个班队都要与其他班队比赛一场，一共进行了120场比赛，这个学校共有多少个班级,3、2008年中山市“光彩杯”中学生足球赛共进行了56场比赛(实行主客场制)，问有多少球队参加比赛,二、面积问题4、利用长为18米的墙的一边，另三边用长为35米的竹篱笆围成一个面积为150平方米的长方形养鸡场。求鸡场的长与宽各多少,25、如图所示要建一个面积为150m的长方形养鸡场,为了节约材料,鸡场的一边靠着原有的一条墙,墙长为25m,另三边用竹篱笆围成,已知篱笆总长为35m.toliftconstruction.3,constructionofroadpavementarenecessary,cleanandfreeofwater,drainagefacilitiesingoodcondition,smoothdrainage.4,FengShuipipeline,communicationfacilities,constructionlightinglayoutreasonable,markclear.5,pointparkconstructionmachineryandequipment,materials,toolsplacedinanorderlymanner,workisfinished,andvehicleappearancelooksclean,fire-fightingequipmentavailable,unobstructed.6,theconstructionofvarioustypesofscaffolding,channel,ladder,guardrails,safetynetsandothersecurityprotectionfacilitiestoimprovereliability,securityIDvisible.7,activelycarryoutdust,toxicity,noisecontrolandarrangetogiveupballastwater.8,livingquartersclean,beautifulenvironment.Constructionsafetyplan15.215.2.1safetyguidelinesthesafetyapproach:people-oriented,fullparticipation,buildasafetyculture,anddynamic.15.2.2safetygoalourcompanyattachesspecialimportancetothisproject,thepromotionofcomprehensivesecurityinthemanagement,constructionmanagement,realization:nopersonaldeathbyaccident;Nomajorconstructionmachineryandequipmentdamageaccidents;Nomajorfireaccidents;Notprimarilyresponsiblefortheoccurrenceofmajoraccidents;Nomajorcollapseandrooffall,oftheaccident;Noenvironmentalpollutionaccidentandoccupationalhealthinjuries,foodpoisoningincidents.Controltheseriousinjuryaccidentrateislessthan(1)求鸡场的长与宽各为多少米?(2)题中的墙长度25m对题目的解起着怎样的作用?6、学校课外生物小组的试验园地是长18米、宽12米的矩形，为便于管理，现要在中间开辟一横两纵三条等宽的小道(如图)，要使种植面积为196平方米，求小道的宽((第3题)ABCDAB7、如图所示,某小区规划在一个长为40米,宽为26米的矩形场地上修建三条同样宽的道路,使其中两条与平行,2AB另一条与垂直,其余部分种草,若使每一块草坪的面积都为144米,求道路的宽度?28、在一幅长80cm，宽50cm的矩形风景画的四周镶一条金色纸边，制成一幅矩形挂图，如果要使整个挂图的面积是5400cm，求金色纸边的宽.29、把一个正方形的一边增加2cm，另一边增加1cm，所得的长方形面积比正方形面积增加14cm,那么原来正方形的边长应是多少,10、如图,从一块长80厘米，宽60厘米的铁片中间截去一个小长方形，使剩下的长方框四周的宽度一样，并且小长方形的面积是原来铁片面积的一半，求这个宽度.toliftconstruction.3,constructionofroadpavementarenecessary,cleanandfreeofwater,drainagefacilitiesingoodcond5ition,smoothdrainage.4,FengShuipipeline,communicationfacilities,constructionlightinglayoutreasonable,markclear.5,pointparkconstructionmachineryandequipment,materials,toolsplacedinanorderlymanner,workisfinished,andvehicleappearancelooksclean,fire-fightingequipmentavailable,unobstructed.6,theconstructionofvarioustypesofscaffolding,channel,ladder,guardrails,safetynetsandothersecurityprotectionfacilitiestoimprovereliability,securityIDvisible.7,activelycarryoutdust,toxicity,noisecontrolandarrangetogiveupballastwater.8,livingquartersclean,beautifulenvironment.Constructionsafetyplan15.215.2.1safetyguidelinesthesafetyapproach:people-oriented,fullparticipation,buildasafetyculture,anddynamic.15.2.2safetygoalourcompanyattachesspecialimportancetothisproject,thepromotionofcomprehensivesecurityinthemanagement,constructionmanagement,realization:nopersonaldeathbyaccident;Nomajorconstructionmachineryandequipmentdamageaccidents;Nomajorfireaccidents;Notprimarilyresponsiblefortheoccurrenceofmajoraccidents;Nomajorcollapseandrooffall,oftheaccident;Noenvironmentalpollutionaccidentandoccupationalhealthinjuries,foodpoisoningincidents.Controltheseriousinjuryaccidentrateislessthan11、墙的一边，再用13米长的铁丝挡三边围成一个面积是20平方米的长方形，问长方形长和宽各是多少才能刚好合适,三、体积问题:12、有一块长25厘米、宽15厘米的长方形铁片，如果在铁成的四个角上截去四个相同的小正方形，然后把四边折起来，做成一个底面积为231平方厘米的无盖长方体盒子，求截去的小正方形的边长应是多少厘米,13、如图22.2.1，一块长和宽分别为60厘米和40厘米的长方形铁皮，要在它的四角截去四个相等的小正方形，折成一个无盖的长方体水槽，使它的底面积为800平方米.求截去正方形的边长.四、增长率问题:14、某化服厂去年4月份生产化肥500吨，因管理不善，5月份的化肥产量减少了10%，从6月份起强化了管理，产量逐月上升，7月份产量达到648吨，求该厂6、7月份的月平均增长率。15、某商厦10月份的营业额是50万元，第四季度的总营业额是182万元，后两个月中，每月的平均增长率是多少,16、某商店四月份电扇的销售量为500台，随着天气的变化，六月份电扇的销售量为720台，问五月份、六月份平均每月电扇销售量的增长率是多少,toliftconstruction.3,constructionofroadpavementarenecessary,cleanandfreeofwater,drainagefacilitiesingoodcondition,smoothdrainage.4,FengShuipipeline,communicationfacilities,constructionlightinglayoutreasonable,markclear.5,pointparkconstructionmachineryandequipment,materials,toolsplacedinanorderlymanner,workisfinished,andvehicleappearancelooksclean,fire-fightingequipmentavailable,unobstructed.6,theconstructionofvarioustypesofscaffolding,channel,ladder,guardrails,safetynetsandothersecurityprotectionfacilitiestoimprovereliability,securityIDvisible.7,activelycarryoutdust,toxicity,noisecontrolandarrangetogiveupballastwater.8,livingquartersclean,beautifulenvironment.Constructionsafetyplan15.215.2.1safetyguidelinesthesafetyapproach:people-oriented,fullparticipation,buildasafetyculture,anddynamic.15.2.2safetygoalourcompanyattachesspecialimportancetothisproject,thepromotionofcomprehensivesecurityinthemanagement,constructionmanagement,realization:nopersonaldeathbyaccident;Nomajorconstructionmachineryandequipmentdamageaccidents;Nomajorfireaccidents;Notprimarilyresponsiblefortheoccurrenceofmajoraccidents;Nomajorcollapseandrooffall,oftheaccident;Noenvironmentalpollutionaccidentandoccupationalhealthinjuries,foodpoisoningincidents.Controltheseriousinjuryaccidentrateislessthan17、某商店4月份销售额为50万元,第二季度的总销售额为182万元,若5、6两个月的月增长率相同,求月增长率.18、某电脑公司2000年的各项经营收入中，经营电脑配件的收入为600万元，占全年经营总收入的40%，该公司预计2002年经营总收入要达到2160万元，且计划从2000年到2002年，每年经营总收入的年增长率相同，问2001年预计经营总收入为多少万元,19、某型号的手机连续两次降价，每个售价由原来的1185元降到了580元.求平均每次降价的百分率。20、某工厂计划经过两年的时间将某种产品的产量从每年144万台提高到169万台，求每年平均增长的百分率。五、利润问题:21、将进货单价为40元的某种商品按50元售出时，就能卖出500个，已知这种商品每个涨价1元，其销售量就减少10个，为了赚得8000元的利润，售价应定为多少,这时应进货多少个,22、将进货单价为40元的商品按50元售出时，就能卖出500个.已知这种商品每个涨价1元，其销售量就减少10个，问为了赚得8000元的利润，售价应定为多少,这时应进货多少个,toliftconstruction.3,constructionofroadpavementarenecessary,cleanandfreeofwater,drainagefacilitiesingoodcond7ition,smoothdrainage.4,FengShuipipeline,communicationfacilities,constructionlightinglayoutreasonable,markclear.5,pointparkconstructionmachineryandequipment,materials,toolsplacedinanorderlymanner,workisfinished,andvehicleappearancelooksclean,fire-fightingequipmentavailable,unobstructed.6,theconstructionofvarioustypesofscaffolding,channel,ladder,guardrails,safetynetsandothersecurityprotectionfacilitiestoimprovereliability,securityIDvisible.7,activelycarryoutdust,toxicity,noisecontrolandarrangetogiveupballastwater.8,livingquartersclean,beautifulenvironment.Constructionsafetyplan15.215.2.1safetyguidelinesthesafetyapproach:people-oriented,fullparticipation,buildasafetyculture,anddynamic.15.2.2safetygoalourcompanyattachesspecialimportancetothisproject,thepromotionofcomprehensivesecurityinthemanagement,constructionmanagement,realization:nopersonaldeathbyaccident;Nomajorconstructionmachineryandequipmentdamageaccidents;Nomajorfireaccidents;Notprimarilyresponsiblefortheoccurrenceofmajoraccidents;Nomajorcollapseandrooffall,oftheaccident;Noenvironmentalpollutionaccidentandoccupationalhealthinjuries,foodpoisoningincidents.Controltheseriousinjuryaccidentrateislessthan23、某商店进了一批服装，进价为每件50元.按每件60元出售时，可销售800件;若单价每提高1元，则其销售量就减少20件.今商店计划获利12000元，问销售单价应定为多少元,此时应进多少件服装,24、某百货商店服装柜在销售中发现:“宝乐”牌童装平均每天可售出20件，每件盈利40元.为了迎接“六?一”国际儿童节，商场决定采取适当的