《2025年名校课堂九年级数学上册人教版》

1. 解下列方程：
(1)$(2x+1)^{2}=9.$
(2)$3(x+1)^{2}-108=0.$
(3)$x^{2}-4x+4=5.$

答案： 1.解：
(1)$2x + 1 = \pm 3.\therefore 2x + 1 = 3$或$2x + 1 = - 3.\therefore x_1 = 1,x_2 = - 2$．
(2)$3(x + 1)^2 = 108,(x + 1)^2 = 36.\therefore x + 1 = \pm 6.\therefore x_1 = 5,x_2 = - 7$．
(3)$(x - 2)^2 = 5.\therefore x - 2 = \pm \sqrt{5}.\therefore x_1 = 2 + \sqrt{5},x_2 = 2 - \sqrt{5}$．

2. 解下列方程：
(1)$x^{2}-2x-99=0.$
(2)$3x^{2}-6x+2=0.$

答案： 2.解：
(1)$x^3 - 2x = 99,x^2 - 2x + 1 = 99 + 1$，即$(x - 1)^2 = 100.\therefore x - 1 = \pm 10.\therefore x_1 = 11,x_2 = - 9$．
(2)$x^2 - 2x = - \frac{2}{3},x^2 - 2x + 1 = - \frac{2}{3} + 1$，即$(x - 1)^2 = \frac{1}{3},\therefore x - 1 = \pm \frac{\sqrt{3}}{3}.\therefore x_1 = 1 + \frac{\sqrt{3}}{3},x_2 = 1 - \frac{\sqrt{3}}{3}$．

3. 解下列方程：
(1)$x(x-2)+x-2=0.$
(2)$9x^{2}-(x-1)^{2}=0.$

答案： 3.解：
(1)$(x + 1)(x - 2) = 0,\therefore x + 1 = 0$或$x - 2 = 0.\therefore x_1 = - 1,x_2 = 2$．
(2)$(3x + x - 1)(3x - x + 1) = 0,(4x - 1)(2x + 1) = 0,\therefore 4x - 1 = 0$或$2x + 1 = 0.\therefore x_1 = \frac{1}{4},x_2 = - \frac{1}{2}$．

4. 解下列方程：
(1)$2x^{2}+2x-1=0.$
(2)$3x^{2}-2x-8=0.$

答案： 4.解：
(1)$\because a = 2,b = 2,c = - 1,\therefore \Delta = b^2 - 4ac = 2^2 - 4 × 2 × ( - 1) = 12 ＞ 0.\therefore x = \frac{-2 \pm \sqrt{12}}{2 × 2} = \frac{-2 \pm 2\sqrt{3}}{4} = \frac{-1 \pm \sqrt{3}}{2}.\therefore x_1 = \frac{-1 + \sqrt{3}}{2},x_2 = \frac{-1 - \sqrt{3}}{2}$．
(2)$\because a = 3,b = - 2,c = - 8,\therefore \Delta = b^2 - 4ac = ( - 2)^2 - 4 × 3 × ( - 8) = 100 ＞ 0.\therefore x = \frac{2 \pm \sqrt{100}}{2 × 3} = \frac{2 \pm 10}{6}.\therefore x_1 = 2,x_2 = \frac{-4}{3}$．