A-Level Mathematics: The Discriminant and Its Applications | A-Level 数学：判别式及其在解题中的应用

📚 A-Level Mathematics: The Discriminant and Its Applications | A-Level 数学：判别式及其在解题中的应用

The discriminant is a compact expression that encodes the most important information about the roots of a quadratic equation. In A-Level Mathematics, mastering the discriminant is essential for solving equations, studying curves, and tackling parameter problems.

判别式是一个简洁的代数式，它包含了二次方程根的最关键信息。在A-Level数学中，掌握判别式是解方程、研究曲线以及处理含参数问题的必备技能。

1. Definition of the Discriminant | 判别式的定义

For a quadratic equation ax²+bx+c=0 with real coefficients and a≠0, the discriminant is defined as the expression inside the square root in the quadratic formula.

对于实数系数二次方程 ax²+bx+c=0（a≠0），判别式定义为求根公式中根号内的表达式。

Δ = b² − 4ac

Using the quadratic formula x = (−b ± √(b²−4ac))/(2a), we see that the sign of Δ determines whether the square root gives a real number, zero, or an imaginary number.

由求根公式 x = (−b ± √(b²−4ac))/(2a) 可知，Δ 的符号决定了根号内是正数、零还是负数，从而决定了根的性质。

2. Nature of the Roots | 根的性质

The value of Δ classifies the roots of ax²+bx+c=0 into three cases.

Δ 的值将二次方程 ax²+bx+c=0 的根分为三种情况。

Δ > 0: two distinct real roots. 方程有两个不相等的实根。
Δ = 0: repeated root, also called a double root. 方程有一个重根，即两个相等的实根。
Δ < 0: no real roots; the two roots are complex conjugates. 方程无实根；两个根为一对共轭复根。

For example, x²−5x+6=0 has Δ=25−24=1>0, so it has two distinct real roots. The equation x²−4x+4=0 has Δ=16−16=0, so it has the repeated root x=2.

例如，x²−5x+6=0 的 Δ=25−24=1>0，因此有两个不同实根；而 x²−4x+4=0 的 Δ=16−16=0，因此有重根 x=2。

3. Discriminant and the Graph of a Quadratic | 判别式与二次函数图像

Since the roots of ax²+bx+c=0 are the x-coordinates of the points where the curve y=ax²+bx+c meets the x-axis, the discriminant describes the number of intersections.

由于 ax²+bx+c=0 的根就是曲线 y=ax²+bx+c 与 x 轴交点的横坐标，所以判别式描述的是抛物线与 x 轴的交点个数。

Δ > 0: the graph crosses the x-axis at two distinct points. 图像与 x 轴有两个不同交点。
Δ = 0: the graph touches the x-axis at exactly one point. 图像与 x 轴相切于一点。
Δ < 0: the graph does not intersect the x-axis. If a>0, the whole curve lies above the x-axis; if a<0, the whole curve lies below it. 图像与 x 轴无交点；若 a>0，整条曲线在 x 轴上方；若 a<0，整条曲线在 x 轴下方。

This visual interpretation is often faster than solving an equation, especially in multiple-choice questions.

这种图像理解通常比直接解方程更快，尤其适用于选择题。

4. Using the Discriminant to Find a Parameter | 利用判别式求参数范围

Discriminant problems often require finding the range of an unknown coefficient.

判别式问题常涉及求未知系数的取值范围。

Example 1: Find the set of values of k for which the equation kx²+2x−3=0 has two distinct real roots.

例1：求 k 的取值范围，使方程 kx²+2x−3=0 有两个不相等的实根。

For a quadratic equation, we need k≠0. Here Δ=4−4(k)(−3)=4+12k. For two distinct real roots, Δ>0, so 4+12k>0, which gives k>−1/3. Combining this with k≠0 gives k∈(−1/3,0)∪(0,∞).

首先要保证方程为二次方程，所以 k≠0。这里 Δ=4−4(k)(−3)=4+12k。若有两个不同实根，需 Δ>0，即 4+12k>0，解得 k>−1/3。再结合 k≠0，得 k∈(−1/3,0)∪(0,∞)。

Example 2: The equation x²−4x+k+1=0 has real roots. Find the maximum value of k.

例2：若方程 x²−4x+k+1=0 有实根，求 k 的最大值。

Since Δ=16−4(k+1)=12−4k and real roots require Δ≥0, we have 12−4k≥0, so k≤3. The maximum value is 3.

由于 Δ=16−4(k+1)=12−4k，有实根需 Δ≥0，即 12−4k≥0，所以 k≤3。因此 k 的最大值为3。

5. Intersections of Lines and Quadratic Curves | 直线与二次曲线的交点

To find the intersections of a straight line y=mx+c and a quadratic curve y=ax²+bx+d, substitute the line into the curve. The resulting quadratic equation has discriminant Δ, which tells us the number of intersection points.

求直线 y=mx+c 与抛物线 y=ax²+bx+d 的交点时，将直线方程代入曲线方程，得到一个关于 x 的二次方程，其判别式 Δ 决定交点的个数。

Δ > 0: two intersection points. 两个交点。
Δ = 0: one intersection point; the line is tangent to the curve. 一个交点，即直线与曲线相切。
Δ < 0: no intersection. 没有交点。

Example: The line y=x+k is tangent to the curve y=x². Find k.

例：直线 y=x+k 与曲线 y=x² 相切，求 k。

Substituting y=x+k into y=x² gives x+k=x², so x²−x−k=0. For tangency, Δ=0. That is, (−1)²−4(1)(−k)=1+4k=0, so k=−1/4.

将 y=x+k 代入 y=x²，得 x+k=x²，即 x²−x−k=0。相切要求 Δ=0，即 (−1)²−4(1)(−k)=1+4k=0，所以 k=−1/4。

6. Discriminant and Inequalities | 判别式与不等式恒成立

The discriminant also helps us decide when a quadratic expression is always positive or always negative.

判别式还能帮助我们判断二次表达式是否恒为正或恒为负。

For q(x)=ax²+bx+c, with a≠0, the following standard results hold:

对于 q(x)=ax²+bx+c（a≠0），以下结论成立：

q(x)>0 for all real x if and only if a>0 and Δ<0. q(x)>0 对一切实数 x 成立，当且仅当 a>0 且 Δ<0。
q(x)≥0 for all real x if and only if a>0 and Δ≤0. q(x)≥0 恒成立，当且仅当 a>0 且 Δ≤0。
q(x)<0 for all real x if and only if a<0 and Δ<0. q(x)<0 恒成立，当且仅当 a<0 且 Δ<0。
q(x)≤0 for all real x if and only if a<0 and Δ≤0. q(x)≤0 恒成立，当且仅当 a<0 且 Δ≤0。

Example: Find the values of k such that x²+kx+4>0 for all real x.

例：求 k 的取值范围，使 x²+kx+4>0 对一切实数 x 成立。

Here a=1>0, so we only need Δ<0. Since Δ=k²−16<0, we get −4

这里 a=1>0，所以只需 Δ<0。由 Δ=k²−16<0，得 −4

7. Maximum and Minimum Values of a Quadratic | 二次函数的最大值与最小值

The discriminant is closely connected to the turning point of a quadratic graph.

判别式与二次函数图像的顶点密切相关。

For y=ax²+bx+c, the x-coordinate of the vertex is x=−b/(2a), and the maximum or minimum value is

对于 y=ax²+bx+c，顶点的横坐标为 x=−b/(2a)，其最大值或最小值为

y = c − b²/(4a) = −Δ/(4a)

If a>0, this is the minimum; if a<0, this is the maximum.

若 a>0，这是最小值；若 a<0，这是最大值。

Example: Find the minimum value of 2x²−8x+5.

例：求 2x²−8x+5 的最小值。

Here Δ=(−8)²−4(2)(5)=64−40=24 and a=2. Therefore the minimum value is −Δ/(4a)=−24/(8)=−3.

这里 Δ=(−8)²−4(2)(5)=64−40=24，a=2。因此最小值为 −Δ/(4a)=−24/8=−3。

8. The Discriminant Method for Rational Functions | 判别式法在有理函数值域中的应用

The discriminant can be used to find the range of certain rational functions by converting the problem into a quadratic equation in x.

对于某些有理函数，可以把求值域的问题转化为关于 x 的二次方程，再利用判别式求值域。

Example: Find the range of the function y=x/(x²+1).

例：求函数 y=x/(x²+1) 的值域。

Rearrange as y(x²+1)=x, which becomes yx²−x+y=0. Since x must be real, this quadratic equation in x must have real solutions, so its discriminant must satisfy Δ≥0.

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A-Level Mathematics: The Discriminant and Its Applications | A-Level 数学：判别式及其在解题中的应用