这部分需要会什么？
- 用因式分解、开方或求根公式解二次方程
- 区分根、零点和横截距
- 用韦达定理求根的和、积及相关表达式
核心概念与公式
将方程整理为 \(ax^2+bx+c=0\)，\(a\ne0\)。若可分解为两个因式之积为零，则至少一个因式为零。一般可用 \(x=(-b\pm\sqrt{b^2-4ac})/(2a)\)，实根要求根号内非负。
若两根为 \(r_1,r_2\)，则 \(r_1+r_2=-b/a\)，\(r_1r_2=c/a\)。函数的零点是输入值 \(r\)，图象的横截距则是点 \((r,0)\)。这些概念相关，但答案形式不同。
做题时怎样使用
- 先把等号一边整理为零，再判断最快的方法
- 使用零乘积性质时，对每个因式分别令其等于零
- 若只问根的和或积，优先使用系数关系，并检查实际情境是否限制根
容易混淆的地方
只有乘积为零时，才能分别令因式为零。\((x-2)(x+3)=4\) 不能直接得到 \(x=2\) 或 \(x=-3\)。
用一个例题串起方法
知识点教学例题求 \(3x^2-5x-2=0\) 的根，并求两根之和。
展开解题思路 ＋
分解为 \((3x+1)(x-2)=0\)，得 \(x=-1/3\) 或 \(x=2\)。两根之和为 \(5/3\)，也可直接用 \(-b/a=5/3\)。
本知识点题库
先独立作答，再查看答案与解析。
二次方程・零点与韦达定理基础
\( x^2-5x-1=0 \)
What is the sum of the solutions to the given equation?
原库参考解析
二次方程・零点与韦达定理挑战
\( x^2+9x+5=0 \)
One solution to the given equation can be written as \( x=\frac{-9+\sqrt{k}}{2} \) , where \( k \) is a constant. What is the value of ?
原库参考解析
二次方程・零点与韦达定理挑战
f (x)=(x-a)(x-b)
The function f is defined by the given equation, where a and b are integer constants. If f (35)>0 , f (38)<0, and f (41)>0 , which of the following could be the value of a+b?
原库参考解析
二次方程・零点与韦达定理挑战
\( 6x^2\left(3x-16\right)\left(3x-u\right)=0 \)
In the given equation, u is a positive constant. The sum of the solution to the equation is 13. What is the value of u?
原库参考解析
二次方程・零点与韦达定理挑战
(x - 21)2= 1
What is the sum of the solutions to the given equation?
原库参考解析
二次方程・零点与韦达定理挑战
56x2 +(56s ＋r )x＋rs =0
ln the given equation, r and s are positive constants. The product of the solutions to the given equation is krs, where k is a constant. What is the value of k?
原库参考解析
二次方程・零点与韦达定理挑战
\( 46x^2+\left(46s+r\right)x+rs=0 \)
In the given equation, r and s are positive constants. The product of the solutions to the given equation is krs, where k is a constant. What is the value of k?
原库参考解析
二次方程・零点与韦达定理挑战
46x2+(46s ＋r)x ＋rs =0
In the given equation, r and s are positive constants. The product of the solutions to the given equation is krs, where k is a constant. What is the value of k?
原库参考解析
二次方程・零点与韦达定理挑战
The quadratic function \( ax^2+100x+c \) has a factor of \( px+q \). \( p \) and \( q \) are positive constants. What is the maximal value of \( ac \)?
原库参考解析
二次方程・零点与韦达定理进阶
y = x2 + 18x + 8
y = x + 8
The graphs of the equations in the given system intersect at the point(x, y) in the xy-plane. What is a possible value of x?
原库参考解析
二次方程・零点与韦达定理挑战
-9x(x＋9) = 90
One solution to the given equation can be written as \( x=-\frac{s+\sqrt{t}}{2} \) , where s and t are positive integers. What is the value of \( \frac{s}{t} \) ?
原库参考解析
二次方程・零点与韦达定理挑战
44x2 + (44b + a)x + ab = 0
In the given equation, a and b are positive constants. The product of the solutions to the given equation is kab , where k is a constant. What is the value of k?
原库参考解析
