25.2.1.1直接开平方法-课件-2026-2027学年人教版数学九年级上册

#人教版九上25.2.1.1直接开平方法专项练习（约1\(\boldsymbol{8}\)00字）##知识要点梳理1.**直接开平方法依据**：平方根的定义。若$x^2=a(a\ge0)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)则$x$是$a$的平方根\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\(x=\)\pm\sqrt{a}$；当$\(a<0\)$时\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)负数没有平方根\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)方程无实数根。2.**适用形式**①$x^2=p\quad(p\ge0)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)解：$\boldsymbol{\(x=\)\pm\sqrt{p}}$；②$\((mx+n)\)^2=p\quad(p\ge0,m\neq0)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)把$\((mx+n)\)$看成整体\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$mx+n=\pm\sqrt{p}$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)再解一次方程。>重点提醒：-右边必须是非负数\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\(p<0\)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)方程没有实数根；-开平方一定要写**正负**\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)容易漏掉负根；-先把平方项单独放在等号左边\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)常数放右边\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)再开方；-适合没有一次项（缺少$bx$）的一元二次方程。3.**解题步骤**①变形：化成平方=常数的形式；②判断：常数$\ge0$才有实数解；③开平方：整体开方\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)带上$\pm$；④解一元一次方程\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)求出两个根；⑤写出方程的解。##一、选择题1.能用直接开平方法求解的方程是（）A.$x^2+2\(\(x=\)0\)$B.$x^2-9=0$C.$x^2-\(\boldsymbol{4}\)x+1=0$D.$x^2+3x-2=0$2.方程$x^2=16$的根是（）A.$\(\(x=\)\(\boldsymbol{4}\)\)$B.$\(x=\)-\(\boldsymbol{4}\)$C.$\(x=\)\pm\(\boldsymbol{4}\)$D.$\(x=\)\pm\(\boldsymbol{8}\)$3.方程$(x-2)^2=0$的解是（）A.$x_1=2,x_2=-2$B.$x_1=x_2=2$C.$\(\(x=\)0\)$D.无实数根\(\boldsymbol{4}\).若方程$(2x+1)^2=k$有实数根\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)则$k$的取值范围是（）A.$\(k>0\)$B.$k\ge0$C.$\(k<0\)$D.$k\le0$5.方程$(x+1)^2=-5$的根的情况是（）A.$x_1=-1+\sqrt5,x_2=-1-\sqrt5$B.$x_1=1+\sqrt5,x_2=1-\sqrt5$C.有两个相等实数根D.无实数根##二、填空题6.若$x^2=5$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)则$\(x=\)$________。7.方程$(x-3)^2=25$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)把________看作整体\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)得$x-3=\pm5$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)解得$x_1=$____\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$x_2=$____。\(\boldsymbol{8}\).方程$3x^2-27=0$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)先移项得$3x^2=27$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)化简$x^2=9$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)根是________。9.若$(2x-1)^2=(x+2)^2$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)直接开平方可得：$2x-1=$________。10.已知$(x+a)^2=b$的一个根是$\(x=\)-5$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\(b=9\)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)则$\(a=\)$________。##三、按要求解方程（直接开平方法）11.$x^2-121=0$12.$\(\boldsymbol{4}\)x^2=\(\boldsymbol{8}\)1$13.$(x+\(\boldsymbol{4}\))^2=16$1\(\boldsymbol{4}\).$(3x-2)^2=\(\boldsymbol{4}\)9$15.$2(x-1)^2-\(\boldsymbol{8}\)=0$16.$(2x+3)^2=(x-1)^2$##四、解答题17.判断下面方程能不能用直接开平方法解\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)如果能\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)写出解；不能\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)说明理由。(1)$x^2+\(\boldsymbol{4}\)=0$(2)$(x-5)^2=12$1\(\boldsymbol{8}\).一个正方形的面积是$(2x-1)^2=16$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)求这个正方形的边长。---#参考答案与解析1.**B**解析：直接开平方法适合只有平方项、无一次项\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)\(\boldsymbol{2或8}\)者整体平方等于常数的形式。$x^2-9=0$化为$x^2=9$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)可以直接开平方。A、C、D都含有一次项\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)不能直接开平方。2.**C**解析：$x^2=16$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$x$是16的平方根\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\(x=\)\pm\sqrt{16}=\pm\(\boldsymbol{4}\)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)两个根\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)不要只写正数。3.**B**解析：$(x-2)^2=0$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)开方$x-2=0$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)两个相等实数根$x_1=x_2=2$。\(\boldsymbol{4}\).**B**解析：平方结果不可能是负数\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)所以$k\ge0$时才有实数根；$\(k<0\)$无实数根。5.**D**解析：左边是平方\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)一定$\ge0$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)右边$-5<0$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)不存在这样的实数\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)方程无实数根。6.$\boldsymbol{\pm\sqrt5}$7.$\boldsymbol{x-3}$；$\boldsymbol{\(\boldsymbol{8}\)}$；$\boldsymbol{-2}$解析：$x-3=5$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\(x=\)\(\boldsymbol{8}\)$；$x-3=-5$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\(x=\)-2$。\(\boldsymbol{8}\).$\boldsymbol{x_1=3,x_2=-3}$9.$\boldsymbol{\pm(x+2)}$解析：两边都是平方\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)开方后左边等于右边\(\boldsymbol{2或8}\)者右边的相反数。10.$\boldsymbol{2\(\boldsymbol{2或8}\)\(\boldsymbol{8}\)}$解析：$\(b=9\)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$(x+a)^2=9$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$x+\(a=\)\pm3$；当$\(x=\)-5$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$-5+\(a=\)3\Rightarrow\(a=\)\(\boldsymbol{8}\)$；$-5+\(a=\)-3\Rightarrow\(a=\)2$。11.$x^2=121$$\(x=\)\pm11$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\boldsymbol{x_1=11,x_2=-11}$12.$x^2=\dfrac{\(\boldsymbol{8}\)1}{\(\boldsymbol{4}\)}$$\(x=\)\pm\dfrac92$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\boldsymbol{x_1=\dfrac92,x_2=-\dfrac92}$13.$x+\(\boldsymbol{4}\)=\pm\(\boldsymbol{4}\)$$x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)\(\(x=\)0\)$；$x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)\(x=\)-\(\boldsymbol{8}\)$$\boldsymbol{x_1=0,x_2=-\(\boldsymbol{8}\)}$1\(\boldsymbol{4}\).$3x-2=\pm7$①$3x-2=7\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)3\(x=\)9\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)\(x=\)3$②$3x-2=-7\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)3\(x=\)-5\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)\(x=\)-\dfrac53$$\boldsymbol{x_1=3,x_2=-\dfrac53}$15.$2(x-1)^2=\(\boldsymbol{8}\)$$(x-1)^2=\(\boldsymbol{4}\)$$x-1=\pm2$$x_1=3\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)x_2=-1$16.$2x+3=\pm(x-1)$①$2x+3=x-1$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$\(x=\)-\(\boldsymbol{4}\)$②$2x+3=-(x-1)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)$2x+3=-x+1\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)3\(x=\)-2\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)\(x=\)-\dfrac23$$\boldsymbol{x_1=-\(\boldsymbol{4}\),x_2=-\dfrac23}$17.(1)$x^2=-\(\boldsymbol{4}\)$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)右边负数\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)**不能**\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)负数没有平方根\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)无实数根。(2)能。$x-5=\pm\sqrt{12}=\pm2\sqrt3$$x_1=5+2\sqrt3\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)x_2=5-2\sqrt3$1\(\boldsymbol{8}\).$(2x-1)^2=16$$2x-1=\pm\(\boldsymbol{4}\)$边长是正数\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)所以$2x-1>0$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)舍去负值：$2x-1=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)2\(x=\)5\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)\(x=\)2.5$；$2x-1=-\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)2\(x=\)-3\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)\(x=\)-1.5$（舍去）正方形边长：$\boldsymbol{\(\boldsymbol{4}\)}$##易错点总结1.开平方**必须带$\boldsymbol{\pm}$**\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)是最常错的地方；2.先把平方项单独留在左边\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)系数要化成1；3.右边是负数→直接判定无实数根；\(\boldsymbol{4}\).实际问题（边长、长度）求出根后\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)要检验\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)负数根舍去；5.形如$(A)^2=(B)^2$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)开方是$\(A=B\)$\(\boldsymbol{2或8}\)$A=-B$两种情况；6.两个根相等的时候\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)要写成$x_1=x_2=a$\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)不能只写一个。全文约17\(\boldsymbol{8}\)0字\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)贴合课本25.2直接开平方法第一课时\(x+\(\boldsymbol{4}\)=\(\boldsymbol{4}\)\(x+\(\boldsymbol{4}\)=-\(\boldsymbol{4}\)\(x_1=3\(x_1=5+2\sqrt3，x_2=5-2\sqrt3\)x_2=-1\)x=-8\)x=0\)难度适合当堂训练。要不要我给你单独出一份**空白试卷版（不带答案）**？人教版（新教材）数学九年级上册25.2.1.1直接开平方法授课教师：.

班

级：

9年级（

）班.

时

间：.

2026/11/3

第二十五章一元二次方程

同学们，认真学习，不只是为了分数。认真听讲、踏实思考，能帮我们把知识学扎实，慢慢拥有分辨是非、独立思考的能力。现在的每一分专注，都会化作以