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
\( x(x+9)-13=0 \)
What are the solutions to the given equation?
What are the solutions to the given equation?
参考答案
二次方程・零点与韦达定理挑战
\( x-a=(x-a)(x-24) \)
Which of the following are solutions to the given equation, where a is a constant and a > 25?
I. a
II. 24
III. 25
Which of the following are solutions to the given equation, where a is a constant and a > 25?
I. a
II. 24
III. 25
参考答案
二次方程・零点与韦达定理挑战
\( y=13x+2 \)
\( y=13x^2+2 \)
Which ordered pair (x, y) is a solution to the given system of equations?
\( y=13x^2+2 \)
Which ordered pair (x, y) is a solution to the given system of equations?
参考答案
二次方程・零点与韦达定理挑战
\( x-3y=12 \)
\( x^2+9y^2=-6xy+16 \)
The solution to the given system of equations is (x, y). Which of the following could be (x, y)?
I: (4, \( -\frac{8}{3} \) )
II: (8, \( -\frac{4}{3} \) )
III: (3, \( −3 \) )
\( x^2+9y^2=-6xy+16 \)
The solution to the given system of equations is (x, y). Which of the following could be (x, y)?
I: (4, \( -\frac{8}{3} \) )
II: (8, \( -\frac{4}{3} \) )
III: (3, \( −3 \) )
参考答案
二次方程・零点与韦达定理进阶
\( y=x^2+22x+4 \)
\( y=x+4 \)
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?
\( y=x+4 \)
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?
参考答案
二次方程・零点与韦达定理进阶
\( x^2-7x+8=0 \)
The solutions to the given equation can be written in the form \( \frac{7±\surd k}{2} \), where k is a constant. What is the value of k?
The solutions to the given equation can be written in the form \( \frac{7±\surd k}{2} \), where k is a constant. What is the value of k?
参考答案
二次方程・零点与韦达定理挑战
The product of a positive number x and the number that is 4 more than x is 117. What is the value of x?
参考答案
二次方程・零点与韦达定理进阶
\( 8(x-6)^2=800 \)
What is the positive solution to the given equation?
What is the positive solution to the given equation?
参考答案
二次方程・零点与韦达定理进阶
\( x^2+27x+1=0 \)
What is one of the solutions to the given equation?
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 h is defined by \( h\left(x\right)=-\sqrt{\left(x^2+bx+c\right)} \) , where b and c are constants. In the xy-plane, the graph of \( y=h(x) \) contains the points \( \left(2,\ 0\right) \) and \( (0,-\sqrt{314}) \) . If \( h(m)=0 \) , what is the greatest possible value of m?
参考答案
二次方程・零点与韦达定理基础
\( (z+2)(z-4)=0 \)
What are all possible solutions to the given equation?
What are all possible solutions to the given equation?
参考答案
二次方程・零点与韦达定理挑战
\( x^2+y+5=5 \)
\( 4x+4-y=0 \)
The solution to the given system of equations is (x, y). What is the value of x?
\( 4x+4-y=0 \)
The solution to the given system of equations is (x, y). What is the value of x?
参考答案
