知识点地图 / 进阶数学
Discriminant, Tangency & Number of Roots
判别式・相切与根的个数
把交点问题化成一元二次方程。先判断判别式的符号，再处理参数范围、整数边界和重根坐标。
这部分需要会什么？
- 用判别式判断实根个数
- 将直线与抛物线的交点问题转为二次方程
- 结合参数限制求相切或无交点条件
核心概念与公式
对真正的二次方程 \(ax^2+bx+c=0\)（\(a\ne0\)），判别式 \(\Delta=b^2-4ac\)。\(\Delta>0\) 有两个不同实根，\(\Delta=0\) 有一个不同实根（重根），\(\Delta<0\) 无实根。
求直线与抛物线交点时，先联立，整理成关于同一个变量的二次方程。相切对应重根，因而判别式为零。若含参数使二次项系数为零，方程可能变成一次方程，必须另外判断。
做题时怎样使用
- 先整理标准式，明确带符号的 \(a,b,c\)
- 把“两个交点、相切、无实根”翻译成判别式的条件
- 解参数条件后检查 \(a\ne0\)，再应用正数、整数等题目限制
容易混淆的地方
判断两条曲线的交点，必须用联立后方程的判别式，不能只看抛物线自身与横轴的交点。
用一个例题串起方法
知识点教学例题直线 \(y=2x+k\) 与抛物线 \(y=x^2+4\) 相切，求 \(k\)。
展开解题思路 ＋
联立得 \(x^2-2x+4-k=0\)。相切要求 \(\Delta=4-4(4-k)=0\)，所以 \(k=3\)。此时方程为 \((x-1)^2=0\)，交点为 \((1,5)\)。
本知识点题库
先独立作答，再查看答案与解析。
判别式・相切与根的个数进阶
\( x^2+\left(\sqrt{k-3}\right)x+33=0 \)
In the given equation, k is a constant. The equation has exactly one real solution. What is the value of k?
In the given equation, k is a constant. The equation has exactly one real solution. What is the value of k?
参考答案
判别式・相切与根的个数挑战
\( 8(x-k)^2-2t+33=0 \)
In the given equation, k and t are integer constants. The equation has no real solution. What is the greatest possible value of t?
In the given equation, k and t are integer constants. The equation has no real solution. What is the greatest possible value of t?
参考答案
判别式・相切与根的个数进阶
\( x^2+nx=4x-29 \)
In the given equation, n is an integer constant. The equation has no real solution. What is the greatest possible value of n?
In the given equation, n is an integer constant. The equation has no real solution. What is the greatest possible value of n?
参考答案
判别式・相切与根的个数挑战
\( y=21x^2+22 \)
\( y=rx \)
The given system of equations, where r is a constant, has two real solutions. If r is a negative integer, what is the greatest possible value of r?
\( y=rx \)
The given system of equations, where r is a constant, has two real solutions. If r is a negative integer, what is the greatest possible value of r?
参考答案
判别式・相切与根的个数挑战
\( -9x(6x+p)=24 \)
In the given equation, p is an integer constant. The equation has no real solution. What is the least possible value of p?
In the given equation, p is an integer constant. The equation has no real solution. What is the least possible value of p?
参考答案
判别式・相切与根的个数挑战
\( (x+k)^2+48=48 \)
In the given equation, k is a constant. How many distinct real solutions does this equation have?
In the given equation, k is a constant. How many distinct real solutions does this equation have?
参考答案
判别式・相切与根的个数挑战
\( x^2+nx=4x-19 \)
In the given equation, n is an integer constant. The equation has no real solution. What is the greatest possible value of n?
In the given equation, n is an integer constant. The equation has no real solution. What is the greatest possible value of n?
参考答案
判别式・相切与根的个数挑战
\( x\left(kx-56\right)=-16 \)
In the given equation, k is an integer constant. If the equation has no real solution, what is the least possible value of k?
原库参考解析
判别式・相切与根的个数进阶
Which quadratic equation has exactly one distinct real solution?
原库参考解析
判别式・相切与根的个数进阶
In the xy-plane, a line with equation 2y = c for some constant c intersects a parabola at exactly one point.If the parabola has equation y = -2x2+9x, what is the value of c?
原库参考解析
判别式・相切与根的个数挑战
64x2 + bx + 25 = 0
In the given equation, b is a constant. For which of the following values of b will the equation have more than one real solution?
原库参考解析
判别式・相切与根的个数挑战
x(kx - 56) = -16
In the given equation, k is an integer constant. If the equation has no real solution, what is the least possible value of k?
原库参考解析
