2.2 一元二次方程的解法第2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2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 1.理解开平方法解一元二次方程的依据是平方根的意义，会用开平方法 解一元二次方程. 2.理解配方法，会用配方法解二次项系数为1的一元二次方程. 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 2 因式分解法解一元二次方程的一般步骤是什么？ 因式分解法解方程的基本步骤： 1.移项：将方程的右边化为0； 2.化积：将方程的左边分解为两个一次因式的乘积； 3.转化：另每一个因式都为0，将解一元二次方程转化为解两个一元一次方程； 4.求解：解这两个一元一次方程，得原方程的解. 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 3 因式分解法： 1.提取公因式法：如x2+ax=0x(x+a)=0 x1 =0, x2 = 2.完全平方公式法：如x2+2ax+a2 =0 (x+a)2=0 x1 =x2 = 3.平方差根式法：如x2a2 =0 (x+a) (xa)=0 x1 = a, x2 = 4.十字相乘法：如x2+(a+b)x+ab =0 (x+a)(x+b)=0 x1 =, x2 = 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 4 解下列方程: (1) x29=0. (2) (x+1)2= 16. 解：(1)将方程的左边分解因式,得(x+3)(x3)=0, 则x+3=0,或x3=0, 解得x1 =3, x2 =3. (2)移项,得(x+1)2 16=0. 将方程的左边分解因式,得(x+14)(x+1+4)=0, 即(x)(x+5)=0, 则x3=0,或x+5=0, 解得x1 =3, x2 =5. 思考：这两个方程是否 还有其它的解法？ 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 5 解下列方程: (1) x29=0. (2) (x+1)2= 16. 解：(1)移项,得x2 =9, 解得x1 =3, x2 =3. (2)由原方程,得x+1=±4, 即x+1=4,或x+1=4, 解得x1 =3, x2 =5. 一般地,对于形如x2 =a(a≥0)的方程,根据平方根的定义,可得 x1 =, x2 =.这种解一元二次方程的方法叫作开平方法. 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 6 开平方法解方程的基本步骤： 1. 将方程转化为x2 =p或(mx+n)2 =p (p≥0) 的形式(即平方项的系数化为1). 2.分情况求解： 当p>0时, x1 =, x2 =或mx1+n= , mx2+n=(再进一步求出 x1和x2的值). 当p=0时, x1 =x2 =或mx1,2+n=(再进一步求出 x的值). 当p<0时, 方程无实数根. 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 7 例1 用开平方法解下列方程: (1) 3x248=0. (2) (2x3)2=7. 解：(1)移项,得3x2= 48, 方程的两边同时除以3,得x2= 16, 解得x1 =, x2 =4. (2)由原方程,得2x3=±， 即2x3= ,或2x3=， 解得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 8 解方程x210x=16. 解：移项,得x210x+16=0, 将方程的左边分解因式,得(x)(x)=0, 则=0,或x=0, 解得x1 =2, x2 =8. 你能将方程x210x=16转化成(x+a)2=b的形式吗?请尝试解这个方程. 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 解方程x210x=16. 解：将方程的两边同时加上25,得x210x+25=9, 即(x)2=9, 则x=3,或x=, 解得x1 =8, x2 =2. 像上面这样,把一元二次方程的左边配成一个完全平方式,右边为一个非负 常数,然后用开平方法求解,这种解一元二次方程的方法叫作配方法. 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 配方法解方程(二次项系数为1)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.配方：方程两边同时加上一次项系数一半的平方. 3.写成的形式. 4.用开平方法求解. 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 11 例2 用配方法解下列方程: (1) x2=1. (2) x2+5x=0. 解：(1)将方程的两边同时加上9,得x2+9=1+9, 即(x+3)2=10, 则x+3=,或x+3=, 解得x1 =3+, x2 = 3. 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 12 例2 用配方法解下列方程: (1) x2=1. (2) x2+5x=0. 解：(2)移项,得x2+5x=, 方程的两边同时加上,得x2+=6+, 即(x+)2= , 即x+= ,或x+=， 解得x1 =1, 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 13 1.方程(x3)2=0的根是(　　) A.x=3 B.x=0 C.x1=x2=3 D.x1=3，x2=3 C 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 2.用配方法解方程x26x8＝0时，配方正确的是( ) A.(x3)2＝17 B.(x3)2＝14　　 C.(x6)2＝44 D.(x3)2＝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 15 3.若方程(x-4)2＝a有实数解，则a的取值范围是( ) A. a≤0 B. a≥0 C. a＞0 D. a＜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 4.下列解方程的结果正确的是( ) A. x2=,解得x=±　　　 B. (x1)2=,解得x1=2,所以x=3 C. x2=7,解得x=±　　　 　　　　 D. 25x2=,解得25x=±所以x=±　　　 　　　　 　　　 C 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 5.一元二次方程x2+x1=0的根是(　　) A.x=1　 B.x= C.x=1+　 D.x= D 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 6.若分式的值为零,则x的值为( ) A. 3　　　　 B.±3 C.. 0　　　　 D.－3 D 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 19 7.规定:a⊗b=b(a+b),如:2⊗3=3×(2+3)=15,若2⊗x=3,则x=　　　　.  8.已知4x2＋8(n＋1)x＋16n是一个关于x的完全平方式，则常数n的值为____． 9.若关于x的一元二次方程ax2＝b (ab＞0)的两根分别是m+1与2m-4，则式子 的值是　　　　.  1或-3 1 10 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 20 10.用开平方法解下列方程: (1)5x280=0;　 (2)(2x+1)2=2. 解： (1)移项,得5x2=80, 方程的两边同除以5,得x2=16, 解得x1=4,x2=4. (2)由原方程,得2x+1=或2x+1=, 解得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 课堂小结 开平方法 解方程的 基本步骤 1.将方程转化为x2 =p或(mx+n)2 =p (p≥0) 的形式(即平方 项的系数化为1). 2.分情况求解： 当p>0时，x1 =, x2 =或mx1+n=， mx2+n= (再进一步求出 x1和x2的值). 当p=0时，x1 =x2 =或mx1,2+n=(再进一步求出 x的值). 当p<0时，方程无实数根. 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 课堂小结 配方法解方程 (二次项系数为 1)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.配方：方程两边同时加上一次项系数一半的平方. 3.写成的形式. 4.用开平方法求解. 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 $