2.2 一元二次方程的解法第4课时 课件2025-2026学年 浙教版数学八年级下册

第2章　一元二次方程 2.2一元二次方程的解法 第4课时　一元二次方程的解法 初中数学浙教版（2024）八年级下册 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 1.经历并理解一元二次方程求根公式的推导过程,会用公式法解一元二次 方程. 2.理解并会计算一元二次方程根的判别式,会用判别式判断一元二次方程 的根的情况. 学习目标 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 2 配方法解方程的基本步骤： 1. 移项：将常数项移到等号右边，含未知数的项移到等号左边. 2.二次项系数化为1 ：如果二次项系数不是1，将方程两边同时除以二次项系数. 3.配方：方程两边同时加上一次项系数一半的平方. 4.写成的形式. 5.用开平方法求解. 用配方法解一元二次方程的基本步骤是什么？ 知识回顾 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 3 思考：任何一个一元二次方程都可以写成一般形式ax2+bx+c=0①,你能否用配方法 得出①的解？ 填空: 方程的两边同除以____,得=0. 移项,得=_____________. 方程的两边同加上_____________，得 +_______= _______. a 新知探究 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 4 即(x+_______)2=， 若≥0, 可得 = ，或 =， ∴,或， 我们也可以简单地表示为. 新知探究 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 5 对于一元二次方程ax2+bx+c=0(a≠0), 如果≥0,那么方程的两个根 为. 这个公式叫作一元二次方程的求根公式.利用求根公式,我们可以由一元二 次方程的系数a,b,c的值,直接求得方程的根.这种解一元二次方程的方法叫 作公式法. 新知探究 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 6 从一元二次方程ax2+bx+c=0(a≠0)的求根公式的推导过程中不难看出,方程的根的 情况由代数式的值来决定.因此叫作一元二次方程的根的判别式, 通常用希腊字母“”表示，即= . 它的值与一元二次方程的根的关系是: >0方程ax2+bx+c=0(a≠0)有两个不相等的实数根; 0方程ax2+bx+c=0(a≠0)有两个相等的实数根; ＜0方程ax2+bx+c=0(a≠0)没有实数根. 思考：方程ax2+bx+c=0(a≠0)的根的情况由什么决定？ 新知探究 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 7 例1 用公式法解下列一 元二次方程: (1) 2x2x=0. (2) 4x2=x. (3) x2=0. 解：(1) 对方程2x2x=0, a=2,b=,c=3, =3=1, ∴ =, ∴ x1 == , x2 =. 例题精讲 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 8 例1 用公式法解下列一 元二次方程: (1) 2x2x=0. (2) 4x2=x. (3) x2=0. 解：(2) 移项，得方程4x2x =0, 则a=4,b=,c=1, ==0， ∴ =, ∴ x1 =x2 =. 例题精讲 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 9 例1 用公式法解下列一 元二次方程: (1) 2x2x=0. (2) 4x2=x. (3) x2=0. 解：(3) 方程的两边同乘4，得3x2x=0, 则a=3,b=,c=, ==88， ∴ = , ∴ x1 = , x2 = . 例题精讲 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 10 公式法解方程的基本步骤： 1.将方程化为一般形式. 2.确定a,b,c的值. 3.计算. 4. >0：方程ax2+bx+c=0(a≠0)有两个不相等的实数根; 0：方程ax2+bx+c=0(a≠0)有两个相等的实数根; ＜0：方程ax2+bx+c=0(a≠0)没有实数根. 新知探究 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 11 例2 解方程：. 解：去括号，得x2x= x2x+4, 化简，得x2x+4= 0, 方程的两边同乘2，得x2x=0, 则a=1,b=,c=8, ==4， ∴ = , ∴ x1 = =4, x2 = . 你能用因式分解法 解该方程吗？ 例题精讲 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 12 例2 解方程：. 解：由原方程，得, 移项，得, ∴ =0, =0 ， ∴ ，或, 解得x1 =4, x2 . 例题精讲 因式分解法 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 13 1.一元二次方程x23x=1中，的值为（ ） A. 5 B.13 C. 13 D. 5 B 随堂练习 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 2.一元二次方程x2-7x-2=0的实数根的情况是（ ） A.有两个不相等的实数根 B.有两个相等的实数根 C.没有实数根 D.不能确定 A 随堂练习 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 15 3.用公式法解方程5x2﹣6＝7x，下列代入公式正确的是（ ） A. B. C. D. B 随堂练习 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 16 4.下列一元二次方程没有实数根的是(　　) A.－2x2＋x＋1＝0 B.x2＋2x＋3＝0 C.x2－x－1＝0 D.x2＋2x－3＝0 B 随堂练习 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 5.用公式法解－x2＋3x＝1时,先求出a,b,c的值,则a,b,c的值依次为(　　) A. －1,3,－1 B. 1,－3,－1 C. －1,－3,－1 D. －1,3,1 A 随堂练习 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 6.用公式法解下列方程： (1)3x2－2x－5＝0. (2)2x2－5x＋3＝0. 解：(1)对方程3x2－2x－5＝0, 有a=3,b=,c= －5, ==64>0， ∴ = , ∴ x1 = = , x2 =. 随堂练习 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 6.用公式法解下列方程： (1)3x2－2x－5＝0. (2)2x2－5x＋3＝0. 解：(2)对方程2x2－5x＋3＝0, 有a=2,b=,c= 3, ==1>0， ∴ = , ∴ x1 = =, x2 = . 随堂练习 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 课堂小结 公式法解方程的基本步骤 1.将方程化为一般形式. 2.确定a,b,c的值. 3.计算. 4. >0：方程ax2+bx+c=0(a≠0)有两个不相等的实数根; 0：方程ax2+bx+c=0(a≠0)有两个相等的实数根; 0：方程ax2+bx+c=0(a≠0)没有实数根. Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS 谢谢 Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS Click to add title Click to add text Second level Third level Fourth level Fifth level OfficePLUS $