一元二次方程根的判别式及根与系数的关系 初三数学主讲教师【-】

一元二次方程根的判别式及根与系数的关系初三数学主讲教师:李绿江A．由解方程引入:

解方程:①x2x10b24ac145②x24x40b24ac16160③2x23x40b24ac9320此方程无实数根,可见,由b24ac的值,可以判断方程根的情况.B．新课:一、判别式ax2bxc0(a0)根的判别式是:b24ac.〔1〕b24ac0方程有两个不相等的实数根〔2〕b24ac0方程有两个相等的实数根〔3〕b24ac0方程没有实数根.2、判别式的应用〔1〕直接判断一元二次方程根的情况;〔2〕由题目给出的一元二次方程根的情况,求出a、b、c中待定系数的值或取值范围.例1不解方程,判断以下方程根的情况.〔1〕2x23x10〔2〕〔3〕5x27x50〔4〕kx2(2k1)xk10(k0)〔1〕2x23x10〔2〕〔3〕5x27x50〔4〕kx2(2k1)xk10(k0)解〔1〕(3)2421980∴方程〔1〕有两个不等的实根.〔2〕∴方程〔2〕有两个相等的实数根.〔3〕(7)2455491000∴方程〔3〕无实数根.〔4〕(2k1)24k(1k)4k24k14k4k28k210〔无论k为何值均有8k210〕∴方程〔4〕有两个不等实根.

今后遇到二次方程马上先由判断一下根的情况这是解题的良好习惯.例2关于x的方程(m2)x22(m1)xm10在以下条件下, 分别求m的非负整数值.〔1〕方程只有一个实数根；〔2〕方程有两个相等的实数根；〔3〕方程有两个不相等的实数根.解:〔1〕当m20即m2时方程为一元一次方程2x30,即m2时,方程只有一个实数根.〔2〕当方程有两个相等的实根时,必须且只需 解出

∴m3时,方程有两个相等的实数根.〔3〕当方程有两个不相等实数根时,必须且只需解出又m是非负整数∴m0或m1小结:使用时必须在a0的前题下.

例3.m取什么值时,关于x的方程2x2(m2)x2m20有两个相等的实数根？并求出这时方程的根.解:∵方程有两个相等的实数根,∴(m2)28(2m2)m212m20(m2)(m10)0∴m12m210当m12时当m210时∴所求m2或m10,方程的根为1或3.例4求证:无论a为任何实数,关于x的方程x2(2a1)xa30总有两个不相等的实数根.证:(2a1)24(a3)4a28a134(a1)29即0无论a为任何实数(a1)20∴4(a1)290∴无论a为任何实数,方程x2(2a1)xa30总有两个不等实根.由例4可知:要说明0常将它配成完全平方式正数.观察下表.方程x1x2x1x2x1x2x22x00220x23x40-41-3-4x25x602356Ⅰ观察两根之和,两根之积与a、b、c的关系;Ⅱ两根之和一次项系数的相反数;两根之积常数项.Ⅲ推广方程ax2bxc0(a0b24ac0)变形为由求根公式与上述观察结果对比,可得到根系关系.二、根系关系1、关于x的方程ax2bxc0(a0,b24ac0)的两根x1、x2与系数a、b、c的关系是:注:应用根系关系的前题是a0且02、根系关系的应用:〔1〕方程的一根,求另一根及字母系数的值.〔2〕两根之间的关系,确定方程中字母系数的值.例5方程的一个根是1,求k及另一根解法一:设方程的另一根为x1∴所求,解法二∵1是方程的根∴∴方程为x21∴所求另一根为引申:假设x21那么对应的方程是什么？即以,1为根的方程为0例6方程x2(m1)x2m10求m满足什么条件时,方程的两根互为相反数？方程的两根互为倒数？方程的一根为零？解:(m1)24(2m1)m26m5①∵两根互为相反数∴两根之和m10,m1,且0∴m1时,方程的两根互为相反数.②∵两根互为倒数

m26m5,∴两根之积2m11m1且0,∴m1时,方程的两根互为倒数.③∵方程一根为0,∴两根之积2m10且0,∴时,方程有一根为零.引申:1、假设ax2bxc0(a00)〔1〕假设两根互为相反数,那么b0;〔2〕假设两根互为倒数,那么ac;〔3〕假设一根为0,那么c0;〔4〕假设一根为1,那么abc0;〔5〕假设一根为1,那么abc0;〔6〕假设a、c异号,方程一定有两个实数根.引申2假设a、b是方程x22x70的两个实数根,求:①a2b2②a23b24b③a35b2b76的值.解:由根系关系ab2,ab7,a272ab272b,①a2b2=(ab)22ab41418.②a23b24b(72a)3(72b)4b2(ab)282(2)2832.③a35b2b76aa25b2b76a(72a)5(72b)b767a2a23511b767a2(72a)3511b7611(ab)497611(2)49765.%z4JeTo%y4JeTo%y3JeTo%y3JeTo%y3IeTo%y3IdTo%y3IdSo%y3IdSn%y3IdSn%y3IdSn$y3IdSn$x3IdSn$x2IdSn$x2HdSn$x2HdSn$x2HcSn$x2HcRn$x2HcRm$x2HcRm$x2HcRm!x2HcRm!w2HcRm!w1HcRm!w1GcRm!w1GcRm!w1GbRm!w1GbQm!w1GbQl!w1GbQl#w1GbQl#w1GbQl#v1GbQl#v0GbQl#v0FbQl#v0FaQl#v0FaQl#v0FaPl#v0FaPk#v0FaPkZv0FaPkZv0FaPkZu0FaPkZu+FaPkZu+EaPkZu+E9PkZu+E9PkZu+E9OkZu+E9OjZu+E9OjYu+E9OjYt+E9OjYt+E9OjYt-E9OjYt-D9OjYt-D8OjYt-D8NjYt-D8NjYt-D8NiYt-D8NiXt-D8NiXs-D8NiXs-D8NiXs)D8NiXs)C8NiXs)C7NiXs)C7MiXs)C7MiXs)C7MhXs)C7MhWs)C7MhWr)C7MhWr(C7MhWr(C7MhWr(B7MhWr(B6MhWr(B6LhWr(B6LgWr(B6LgWr(B6LgVr(B6LgVq(B6LgVq*B6LgVq*A6LgVq*A6LgVq*A5LgVq*A5KgVq*A5KfVq*A5KfVq*A5KfUq*A5KfUp*A5KfUp&A5KfUp&z5KfUp&z5KfUp&z4KfUp&z4JfUp&z4JeUp&z4JeTp&z4JeTp&z4JeTo&z4JeTo%z4JeTo%y4JeTo%y3JeTo%y3JeTo%y3IeTo%y3IdTo%y3IdSo%y3IdSo%y3IdSn%y3IdSn$y3IdSn$x3IdSn$x2IdSn$x2IdSn$x2HdSn$x2HcSn$x2HcRn$x2HcRm$x2HcRm$x2HcRm!x2HcRm!w2HcRm!w1HcRm!w1GcRm!w1GcRm!w1GbRm!w1GbQm!w1GbQl!w1GbQl!w1GbQl#w1GbQl#v1GbQl#v0GbQl#v0FbQl#v0FbQl#v0FaQl#v0FaPl#v0FaPk#v0FaPkZv0FaPkZv0FaPkZu0FaPkZu+FaPkZu+EaPkZu+E9PkZu+E9PkZu+E9OkZu+E9OjZu+E9OjYu+E9OjYt+E9OjYt+E9OjYt-E9OjYt-D9OjYt-D8OjYt-D8OjYt-D8NjYt-D8NiYt-D8NiXt-D8NiXs-D8NiXs-D8NiXs)D8NiXs)C8NiXs)C7NiXs)C7MiXs)C7MiXs)C7MhXs)C7MhWs)C7MhWr)C7MhWr(C7MhWr(C7MhWr(B7MhWr(B6MhWr(B6LhWr(B6LhWr(B6LgWr(B6LgVr(B6LgVq(B6LgVq*B6LgVq*B6LgVq*A6LgVq*A5LgVq*A5KgVq*A5KfVq*A5KfVq*A5KfUq*A5KfUp*A5KfUp&A5KfUp&z5KfUp&z5KfUp&z4KfUp&z4JfUp&z4JeUp&z4JeUp&z4JeTp&z4JeTo&z4JeTo%z4JeTo%y4JeTo%y4JeTo%y3JeTo%y3IeTo%y3IdTo%y3IdSo%y3IdSo%y3IdSn%y3IdSn$y3IdSn$x3IdSn$x2IdSn$x2IdSn$x2HdSn$x2HcSn$x2HcRn$x2HcRm$x2HcRm$x2HcRm!x2HcRm!w2HcRm!w1HcRm!w1HcRm!w1GcRp&z4KfUp&z4JfUp&z4JeUp&z4JeTp&z4JeTo&z4JeTo&z4JeTo%z4JeTo%y4JeTo%y3JeTo%y3JeTo%y3IeTo%y3IdTo%y3IdSo%y3IdSn%y3IdSn%y3IdSn$y3IdSn$x3IdSn$x2IdSn$x2HdSn$x2HdSn$x2HcSn$x2HcRn$x2HcRm$x2HcRm!x2HcRm!x2HcRm!w2HcRm!w1HcRm!w1GcRm!w1GbRm!w1GbRm!w1GbQm!w1GbQl!w1GbQl#w1GbQl#w1GbQl#v1GbQl#v0GbQl#v0FbQl#v0FaQl#v0FaQl#v0FaPl#v0FaPk#v0FaPkZv0FaPkZu0FaPkZu0FaPkZu+FaPkZu+EaPkZu+E9PkZu+E9OkZu+E9OkZu+E9OjZu+E9OjYu+E9OjYt+E9OjYt+E9OjYt-E9OjYt-D9OjYt-D8OjYt-D8NjYt-D8NjYt-D8NiYt-D8NiXt-D8NiXs-D8NiXs)D8NiXs)D8NiXs)C8NiXs)C7NiXs)C7MiXs)C7MhXs)C7MhXs)C7MhWs)C7MhWr)C7MhWr(C7MhWr(C7MhWr(B7MhWr(B6MhWr(B6LhWr(B6LgWr(B6LgWr(B6LgVr(B6LgVq(B6LgVq*B6LgVq*A6LgVq*A6LgVq*A5LgVq*A5KgVq*A5KfVq*A5KfUq*A5KfUq*A5KfUp*A5KfUp&A5KfUp&z5KfUp&z4KfUp&z4KfUp&z4JfUp&z4JeUp&z4JeTp&z4JeTp&z4JeTo&z4JeTo%z4JeTo%y4JeTo%y3JeTo%y3JeTo%y3IeTo%y3IdTo%y3IdSo%y3IdSn%y3IdSn%y3IdSn$y3IdSn$x3IdSn$x2IdSn$x2HdSn$x2HdSn$x2HcSn$x2HcRn$x2HcRm$x2HcRm$x2HcRm!x2HcRm!w2HcRm!w1HcRm!w1GcRm!w1GcRm!w1GbRm!w1GbQm!w1GbQl!w1GbQl#w1GbQl#w1GbQl#v1GbQl#v0GbQl#v0FbQl%y3JeTo%y3IeTo%y3IeTo%y3IdTo%y3IdSo%y3IdSn%y3IdSn$y3IdSn$y3IdSn$x3IdSn$x2IdSn$x2HdSn$x2HcSn$x2HcSn$x2HcRn$x2HcRm$x2HcRm!x2HcRm!w2HcRm!w2HcRm!w1HcRm!w1GcRm!w1GbRm!w1GbRm!w1GbQm!w1GbQl!w1GbQl#w1GbQl#v1GbQl#v1GbQl#v0GbQl#v0FbQl#v0FaQl#v0FaPl#v0FaPl#v0FaPk#v0FaPkZv0FaPkZu0FaPkZu+FaPkZu+FaPkZu+EaPkZu+E9PkZu+E9OkZu+E9OkZu+E9OjZu+E9OjYu+E9OjYt+E9OjYt-E9OjYt-E9OjYt-D9OjYt-D8OjYt-D8NjYt-D8NiYt-D8NiYt-D8NiXt-D8NiXs-D8NiXs)D8NiXs)C8NiXs)C8NiXs)C7NiXs)C7MiXs)C7MhXs)C7MhXs)C7MhWs)C7MhWr)C7MhWr(C7MhWr(B7MhWr(B7MhWr(B6MhWr(B6LhWr(B6LgWr(B6LgVr(B6LgVr(B6LgVq(B6LgVq*B6LgVq*A6LgVq*A5LgVq*A5LgVq*A5KgVq*A5KfVq*A5KfUq*A5KfUp*A5KfUp*A5KfUp&A5KfUp&z5KfUp&z4KfUp&z4KfUp&z4JfUp&z4JeUp&z4JeTp&z4JeTo&z4JeTo&z4JeTo%z4JeTo%y4JeTo%y3JeWr(B7MhWr(B6MhWr(B6MhWr(B6LhWr(B6LgWr(B6LgVr(B6LgVq(B6LgVq(B6LgVq*B6LgVq*A6LgVq*A5LgVq*A5KgVq*A5KgVq*A5KfVq*A5KfUq*A5KfUp*A5KfUp&A5KfUp&A5KfUp&z5KfUp&z4KfUp&z4JfUp&z4JfUp&z4JeUp&z4JeTp&z4JeTo&z4JeTo%z4JeTo%z4JeTo%y4JeTo%y3JeTo%y3IeTo%y3IdTo%y3IdTo%y3IdSo%y3IdSn%y3IdSn$y3IdSn$x3IdSn$x3IdSn$x2IdSn$x2HdSn$x2HcSn$x2HcRn$x2HcRn$x2HcRm$x2HcRm!x2HcRm!w2HcRm!w2HcRm!w1HcRm!w1GcRm!w1GbRm!w1GbQm!w1GbQm!w1GbQl!w1GbQl#w1GbQl#v1GbQl#v0GbQl#v0GbQl#v0FbQl#v0FaQl#v0FaPl#v0FaPk#v0FaPk#v0FaPkZv0FaPkZu0FaPkZu+FaPkZu+FaPkZu+EaPkZu+E9PkZu+E9OkZu+E9OjZu+E9OjZu+E9OjYu+E9OjYt+E9OjYt-E9OjYt-D9OjYt-D9OjYt-D8OjYt-D8NjYt-D8NiYt-D8NiXt-D8NiXt-D8NiXs-D8NiXs)D8NiXs)C8NiXs)C7NiXs)C7NiXs)C7MiXs)C7MhXs)C7MhWs)C7Mk#v0FaPkZv0FaPkZv0FaPkZu0FaPkZu+FaPkZu+EaPkZu+E9PkZu+E9PkZu+E9OkZu+E9OjZu+E9OjYu+E9OjYu+E9OjYt+E9OjYt-E9OjYt-D9OjYt-D8OjYt-D8OjYt-D8NjYt-D8NiYt-D8NiXt-D8NiXs-D8NiXs-D8NiXs)D8NiXs)C8NiXs)C7NiXs)C