一元二次方程根的判別式及根與系數(shù)的關(guān)系 初三數(shù)學(xué)主講教師【-】

一元二次方程根的判別式及根與系數(shù)的關(guān)系初三數(shù)學(xué)主講教師:李綠江A．由解方程引入:

解方程:①x2x10b24ac145②x24x40b24ac16160③2x23x40b24ac9320此方程無實(shí)數(shù)根,可見,由b24ac的值,可以判斷方程根的情況.B．新課:一、判別式ax2bxc0(a0)根的判別式是:b24ac.〔1〕b24ac0方程有兩個(gè)不相等的實(shí)數(shù)根〔2〕b24ac0方程有兩個(gè)相等的實(shí)數(shù)根〔3〕b24ac0方程沒有實(shí)數(shù)根.2、判別式的應(yīng)用〔1〕直接判斷一元二次方程根的情況;〔2〕由題目給出的一元二次方程根的情況,求出a、b、c中待定系數(shù)的值或取值范圍.例1不解方程,判斷以下方程根的情況.〔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〕有兩個(gè)不等的實(shí)根.〔2〕∴方程〔2〕有兩個(gè)相等的實(shí)數(shù)根.〔3〕(7)2455491000∴方程〔3〕無實(shí)數(shù)根.〔4〕(2k1)24k(1k)4k24k14k4k28k210〔無論k為何值均有8k210〕∴方程〔4〕有兩個(gè)不等實(shí)根.

今后遇到二次方程馬上先由判斷一下根的情況這是解題的良好習(xí)慣.例2關(guān)于x的方程(m2)x22(m1)xm10在以下條件下, 分別求m的非負(fù)整數(shù)值.〔1〕方程只有一個(gè)實(shí)數(shù)根；〔2〕方程有兩個(gè)相等的實(shí)數(shù)根；〔3〕方程有兩個(gè)不相等的實(shí)數(shù)根.解:〔1〕當(dāng)m20即m2時(shí)方程為一元一次方程2x30,即m2時(shí),方程只有一個(gè)實(shí)數(shù)根.〔2〕當(dāng)方程有兩個(gè)相等的實(shí)根時(shí),必須且只需 解出

∴m3時(shí),方程有兩個(gè)相等的實(shí)數(shù)根.〔3〕當(dāng)方程有兩個(gè)不相等實(shí)數(shù)根時(shí),必須且只需解出又m是非負(fù)整數(shù)∴m0或m1小結(jié):使用時(shí)必須在a0的前題下.

例3.m取什么值時(shí),關(guān)于x的方程2x2(m2)x2m20有兩個(gè)相等的實(shí)數(shù)根？并求出這時(shí)方程的根.解:∵方程有兩個(gè)相等的實(shí)數(shù)根,∴(m2)28(2m2)m212m20(m2)(m10)0∴m12m210當(dāng)m12時(shí)當(dāng)m210時(shí)∴所求m2或m10,方程的根為1或3.例4求證:無論a為任何實(shí)數(shù),關(guān)于x的方程x2(2a1)xa30總有兩個(gè)不相等的實(shí)數(shù)根.證:(2a1)24(a3)4a28a134(a1)29即0無論a為任何實(shí)數(shù)(a1)20∴4(a1)290∴無論a為任何實(shí)數(shù),方程x2(2a1)xa30總有兩個(gè)不等實(shí)根.由例4可知:要說明0常將它配成完全平方式正數(shù).觀察下表.方程x1x2x1x2x1x2x22x00220x23x40-41-3-4x25x602356Ⅰ觀察兩根之和,兩根之積與a、b、c的關(guān)系;Ⅱ兩根之和一次項(xiàng)系數(shù)的相反數(shù);兩根之積常數(shù)項(xiàng).Ⅲ推廣方程ax2bxc0(a0b24ac0)變形為由求根公式與上述觀察結(jié)果對比,可得到根系關(guān)系.二、根系關(guān)系1、關(guān)于x的方程ax2bxc0(a0,b24ac0)的兩根x1、x2與系數(shù)a、b、c的關(guān)系是:注:應(yīng)用根系關(guān)系的前題是a0且02、根系關(guān)系的應(yīng)用:〔1〕方程的一根,求另一根及字母系數(shù)的值.〔2〕兩根之間的關(guān)系,確定方程中字母系數(shù)的值.例5方程的一個(gè)根是1,求k及另一根解法一:設(shè)方程的另一根為x1∴所求,解法二∵1是方程的根∴∴方程為x21∴所求另一根為引申:假設(shè)x21那么對應(yīng)的方程是什么？即以,1為根的方程為0例6方程x2(m1)x2m10求m滿足什么條件時(shí),方程的兩根互為相反數(shù)？方程的兩根互為倒數(shù)？方程的一根為零？解:(m1)24(2m1)m26m5①∵兩根互為相反數(shù)∴兩根之和m10,m1,且0∴m1時(shí),方程的兩根互為相反數(shù).②∵兩根互為倒數(shù)

m26m5,∴兩根之積2m11m1且0,∴m1時(shí),方程的兩根互為倒數(shù).③∵方程一根為0,∴兩根之積2m10且0,∴時(shí),方程有一根為零.引申:1、假設(shè)ax2bxc0(a00)〔1〕假設(shè)兩根互為相反數(shù),那么b0;〔2〕假設(shè)兩根互為倒數(shù),那么ac;〔3〕假設(shè)一根為0,那么c0;〔4〕假設(shè)一根為1,那么abc0;〔5〕假設(shè)一根為1,那么abc0;〔6〕假設(shè)a、c異號,方程一定有兩個(gè)實(shí)數(shù)根.引申2假設(shè)a、b是方程x22x70的兩個(gè)實(shí)數(shù)根,求:①a2b2②a23b24b③a35b2b76的值.解:由根系關(guān)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