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・零点与韦达定理进阶
The product of a positive number x and the number that is 5 less than x is equal to 104. What is the value of x?
原库参考解析
二次方程・零点与韦达定理进阶
The product of a positive number x and the number that is 4 more than x is 60. What is the value of x?
原库参考解析
二次方程・零点与韦达定理进阶
\( x^2-12x+8=0 \)
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
\( (x+5)(x-11)=0 \)
What is the positive solution to the given equation?
原库参考解析
二次方程・零点与韦达定理进阶
\( x^2+5=(18)^2+5 \)
What is a solution to the given equation?
原库参考解析
二次方程・零点与韦达定理挑战
The function h is defined by \( h(x)=-\sqrt{x^2+bx+c} \) where b and c are constants. In the xy-plane, the graph of \( y=h(x) \) contains the points \( (2,\ 0) \) and \( (0,-\sqrt{390}) \) . If \( h(m)=0 \), what is the greatest possible value of m?
参考答案
二次方程・零点与韦达定理进阶
In the xy-plane, the graph of \( y=4x \) intersects the graph of \( y=x^2−77 \) at two points. What is the sum of the x-coordinates of the two points of intersection?
参考答案
二次方程・零点与韦达定理进阶
\( x^2-2x=29 \)
What is one of the solutions to the given equation?
参考答案
二次方程・零点与韦达定理挑战
The function g is defined by \( g(x)=(x+17)(t−x) \), where t is a constant. In the xy-plane, the graph of \( y=g(x) \) passes through the point (26, 0). What is the value of \( g(0) \)?
参考答案
二次方程・零点与韦达定理进阶
\( 49x^2-18=7 \)
Which of the following is a solution to the given equation?
Which of the following is a solution to the given equation?
参考答案
二次方程・零点与韦达定理进阶
The function g is defined by \( g(x)=(x+17)(t-x) \) , where t is a constant. In the xy-plane, the graph of \( y=g(x) \) passes through the point \( (24,\ 0) \). What is the value of \( g(0) \) ?
参考答案
二次方程・零点与韦达定理挑战
\( 48x^2+48x=-3 \)
One solution to the given equation can be written as \( x=\frac{-6-k\sqrt{3}}{12} \), where k is a constant. What is the value of k?
One solution to the given equation can be written as \( x=\frac{-6-k\sqrt{3}}{12} \), where k is a constant. What is the value of k?
参考答案
