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1 [2024·浙江杭州期末]方程x(x - 2)=0的两个根的和是 ( )
A. -2
B.0
C.2
D.4
A. -2
B.0
C.2
D.4
答案：
C
2 [2024·广西南宁月考]方程x² - 6x = 0的解是________.
答案：
$x_1 = 0,x_2 = 6$
3 用因式分解法解下列方程：
(1)[2023·浙江杭州滨江期末]x² = 2x;
(2)2x(x - 3)+x = 3.
(1)[2023·浙江杭州滨江期末]x² = 2x;
(2)2x(x - 3)+x = 3.
答案：
解
(1)将方程整理,得$x^2 - 2x = 0$.
将方程的左边分解因式,得$x(x - 2)=0$,
则$x = 0$,或$x - 2 = 0$,解得$x_1 = 0,x_2 = 2$.
(2)将方程整理,得$2x(x - 3)+(x - 3)=0$.
将方程的左边分解因式,得$(x - 3)(2x + 1)=0$,
则$x - 3 = 0$,或$2x + 1 = 0$,解得$x_1 = 3,x_2 = -\frac{1}{2}$.
(1)将方程整理,得$x^2 - 2x = 0$.
将方程的左边分解因式,得$x(x - 2)=0$,
则$x = 0$,或$x - 2 = 0$,解得$x_1 = 0,x_2 = 2$.
(2)将方程整理,得$2x(x - 3)+(x - 3)=0$.
将方程的左边分解因式,得$(x - 3)(2x + 1)=0$,
则$x - 3 = 0$,或$2x + 1 = 0$,解得$x_1 = 3,x_2 = -\frac{1}{2}$.
4 新题型 阅读辨析题[2023·浙江杭州钱塘三模]圆圆与方方两位同学解方程3(x - 3)=(x - 3)²的过程如下框：
|圆圆：|方方：|
|方程的两边同除以(x - 3),得3 = x - 3.解得x = 6.|移项,得3(x - 3)-(x - 3)² = 0.提取公因式,得(x - 3)(3 - x - 3)=0,则x - 3 = 0,或3 - x - 3 = 0.解得x₁ = 3,x₂ = 0.|
他们的解法是否正确？若错误,请写出你认为正确的解答过程.
|圆圆：|方方：|
|方程的两边同除以(x - 3),得3 = x - 3.解得x = 6.|移项,得3(x - 3)-(x - 3)² = 0.提取公因式,得(x - 3)(3 - x - 3)=0,则x - 3 = 0,或3 - x - 3 = 0.解得x₁ = 3,x₂ = 0.|
他们的解法是否正确？若错误,请写出你认为正确的解答过程.
答案：
解 他们的解法错误.
正确的解法:
移项,得$3(x - 3)-(x - 3)^2 = 0$.
将方程的左边分解因式,得$(x - 3)[3-(x - 3)] = 0$,
则$x - 3 = 0$,或$3-(x - 3)=0$,解得$x_1 = 3,x_2 = 6$.
正确的解法:
移项,得$3(x - 3)-(x - 3)^2 = 0$.
将方程的左边分解因式,得$(x - 3)[3-(x - 3)] = 0$,
则$x - 3 = 0$,或$3-(x - 3)=0$,解得$x_1 = 3,x_2 = 6$.
5 若代数式y² + 7y + 6的值与y - 3的值相等,则y等于 ( )
A. -3
B.3
C. -1或9
D. -9或1
A. -3
B.3
C. -1或9
D. -9或1
答案：
A
6 方程9(x + 1)²-(1 - 2x)² = 0的根为________.
答案：
$x_1 = - 4,x_2 = -\frac{2}{5}$
7 方程x(x - 6)+9 = 0的解是________.
答案：
$x_1 = x_2 = 3$
8 链教材P31作业题第2题改编 用因式分解法解下列方程：
(1)4x² + 1 = 4x;
(2)(3x - 4)² = (4x - 3)²;
(3)2x² - 2$\sqrt{2}$x + 1 = 0.
(1)4x² + 1 = 4x;
(2)(3x - 4)² = (4x - 3)²;
(3)2x² - 2$\sqrt{2}$x + 1 = 0.
答案：
解
(1)将方程整理,得$4x^2 - 4x + 1 = 0$.
将方程的左边分解因式,得$(2x - 1)^2 = 0$,
$\therefore 2x - 1 = 0$,解得$x_1 = x_2 = \frac{1}{2}$.
(2)$\because(3x - 4)^2=(4x - 3)^2,\therefore 3x - 4 = \pm(4x - 3)$,
$\therefore 3x - 4 = 4x - 3$,或$3x - 4 = - 4x + 3$,解得$x_1 = - 1,x_2 = 1$.
(3)将方程的左边分解因式,得$(\sqrt{2}x - 1)^2 = 0$,
$\therefore\sqrt{2}x - 1 = 0$,解得$x_1 = x_2 = \frac{\sqrt{2}}{2}$.
(1)将方程整理,得$4x^2 - 4x + 1 = 0$.
将方程的左边分解因式,得$(2x - 1)^2 = 0$,
$\therefore 2x - 1 = 0$,解得$x_1 = x_2 = \frac{1}{2}$.
(2)$\because(3x - 4)^2=(4x - 3)^2,\therefore 3x - 4 = \pm(4x - 3)$,
$\therefore 3x - 4 = 4x - 3$,或$3x - 4 = - 4x + 3$,解得$x_1 = - 1,x_2 = 1$.
(3)将方程的左边分解因式,得$(\sqrt{2}x - 1)^2 = 0$,
$\therefore\sqrt{2}x - 1 = 0$,解得$x_1 = x_2 = \frac{\sqrt{2}}{2}$.
9 [2024·浙江杭州一模]方程(x - 2)² = 2x(x - 2)的解是 ( )
A.x₁ = 2,x₂ = 1
B.x₁ = 2,x₂ = -2
C.x₁ = 2,x₂ = 0
D.x₁ = 2,x₂ = -1
A.x₁ = 2,x₂ = 1
B.x₁ = 2,x₂ = -2
C.x₁ = 2,x₂ = 0
D.x₁ = 2,x₂ = -1
答案：
B
10 等腰三角形的腰长为2,底边长是方程x² - 8x + 15 = 0的根,则三角形的周长为 ( )
A.7
B.9
C.10
D.7或9
A.7
B.9
C.10
D.7或9
答案：
A
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