📚 The Discriminant | 判别式
4ac tells you immediately whether the equation has two distinct real roots, one repeated real root, or no real roots - without factorising and without applying the quadratic formula. The same idea extends to the intersections of graphs, to quadratic inequalities, and to the hidden conditions that appear in modelling problems.
判别式是 A-Level 二次函数内容中一个简短却十分有力的工具。对于写成标准形式 ax² + bx + c = 0 的任意二次方程，b² − 4ac 的值能让你立刻判断：方程有两个不同实根、一个重根，还是没有实根——既不需要因式分解，也不需要套用求根公式。这一思想同样适用于图像交点、二次不等式，以及建模问题中出现的隐藏条件。
1. What Is the Discriminant? | 什么是判别式
, where a, b and c are real coefficients and a ≠ 0, the discriminant is the expression underneath the square root in the quadratic formula. It is usually denoted by the Greek capital letter Δ (delta).
对于二次方程 ax² + bx + c = 0，其中 a、b、c 为实系数且 a ≠ 0，判别式就是求根公式中平方根号下的表达式，通常用希腊大写字母 Δ（delta）表示。
Δ = b² − 4ac
Because Δ is built only from the coefficients, it depends on the shape of the quadratic and not on the unknown x. Once the equation is in standard form you can evaluate Δ in a single line and read off the nature of the roots.
由于 Δ 只取决于系数，它只与二次函数的形状有关，而与未知数 x 无关。只要方程写成标准形式，你就能一行算出 Δ，直接读出根的性质。
nd, the coefficient a must be non-zero; if a = 0 the equation is linear and the discriminant has no meaning.
使用前有两点提醒。第一，方程必须先整理成标准形式，即所有项移到一边、另一边为零。第二，系数 a 必须不为零；若 a = 0，方程是一次方程，判别式就没有意义了。
2. Derivation from the Quadratic Formula | 从求根公式推导
y the expression below, and the discriminant is simply the part that sits inside the square root.
求根公式表明 ax² + bx + c = 0 的解由下式给出，而判别式正是位于平方根号内部的那一部分。
x = (−b ± √b² − 4ac) / (2a)
. If it is zero, the ± sign makes no difference and both branches give the same value x = −b / (2a), which is a repeated real root. If it is negative, the square root of a negative number is not real, so the equation has no real roots.
分子中包含 b² − 4ac 的平方根。如果 b² − 4ac 为正，平方根是实数，± 号会给出两个不同的值。如果它等于零，± 号就不再起作用，两种取法都得到同一个值 x = −b / (2a)，即一个重根。如果它为负，负数的平方根不是实数，因此方程没有实根。
You can also reach the same conclusion by completing the square, which is where the expression really comes from.
你也可以通过配方得到同样的结论，事实上判别式正是从这里诞生的。
(x + b / (2a))² = (b² − 4ac) / (4a²)
t-hand side can be square-rooted in two ways, zero gives one solution, and a negative right-hand side cannot occur for real x.
因为 a ≠ 0，分母 4a² 恒为正，所以右端的符号完全由分子 b² − 4ac = Δ 决定。右端为正时可以开方得到两个值，右端为零时只得到一个解，而右端为负时，对实数 x 而言不可能成立。
3. The Three Cases | 三种情形
There are exactly three possibilities, and examiners expect you to state which one applies and why.
总共只有三种可能，考官希望你明确指出属于哪一种，并说明理由。
| Value of Δ | Δ 的取值 | Nature of roots | 根的性质 | Graph meets x-axis | 图像与 x 轴 |
| Δ > 0 | Two distinct real roots | 两个不同实根 | Two points | 两个交点 |
| Δ = 0 | One repeated real root | 一个重根 | Tangent, one point | 相切，一个交点 |
| Δ < 0 | No real roots | 没有实根 | No intersection | 没有交点 |
mainly to Further Maths, but the phrase no real roots is expected in every specification.
当 Δ < 0 时，两个根是一对共轭复数，可写成 p ± qi 的形式。这一写法主要属于进阶数学（Further Maths），但“没有实根”这一结论是所有考试局都要求的表述。
It means there is exactly one solution, counted twice, and that solution is x = −b / (2a).
请特别注意，Δ = 0 并不意味着方程无解，而是说明恰有一个解（按重数计为两个），该解为 x = −b / (2a)。
4. Reading the Graph | 图像解读
meets the x-axis, because those points are exactly the real solutions of ax² + bx + c = 0.
判别式统计的正是曲线 y = ax² + bx + c 与 x 轴相交的次数，因为这些交点恰好对应方程 ax² + bx + c = 0 的实数解。
A positive discriminant gives two crossings, a zero discriminant gives a single touching point at the vertex, and a negative discriminant gives a curve that floats entirely above or entirely below the axis.
判别式为正时曲线与 x 轴有两个交点；判别式为零时曲线在顶点处与 x 轴相切于一点；判别式为负时，整条曲线完全浮在 x 轴上方或完全沉在下方。
A frequent error is to confuse the sign of Δ with the sign of a. The sign of a decides whether the parabola opens upwards (a > 0) or downwards (a < 0); the sign of Δ decides whether it crosses the axis. The two are independent, so all four combinations are possible.
常见错误是把 Δ 的符号与 a 的符号混为一谈。a 的符号决定抛物线开口向上（a > 0）还是向下（a < 0）；Δ 的符号决定它是否与 x 轴相交。两者彼此独立，因此四种组合都可能出现。
- a > 0, Δ < 0: opens upwards, always positive | 开口向上，恒为正
- a > 0, Δ = 0: opens upwards, touches the axis once | 开口向上，与 x 轴相切一次
- a < 0, Δ < 0: opens downwards, always negative | 开口向下，恒为负
- a < 0, Δ > 0: opens downwards, crosses twice | 开口向下，与 x 轴交于两点
5. Worked Example: Nature of Roots | 例题：判断根的性质
Example 1. Determine the nature of the roots of 2x² − 4x + 5 = 0.
例 1. 判断方程 2x² − 4x + 5 = 0 的根的性质。
ation has no real roots, and the graph of y = 2x² − 4x + 5 lies entirely above the x-axis.
这里 a = 2，b = −4，c = 5，所以 Δ = (−4)² − 4 × 2 × 5 = 16 − 40 = −24。由于 Δ < 0，方程没有实根，曲线 y = 2x² − 4x + 5 完全位于 x 轴上方。
Example 2. Determine the nature of the roots of x² − 6x + 9 = 0.
例 2. 判断方程 x² − 6x + 9 = 0 的根的性质。
x = 6 / 2 = 3. Indeed x² − 6x + 9 = (x − 3)², and the curve touches the x-axis at (3, 0).
这里 a = 1，b = −6，c = 9，所以 Δ = 36 − 36 = 0。方程有一个重根 x = 6 / 2 = 3。事实上 x² − 6x + 9 = (x − 3)²，曲线在点 (3, 0) 处与 x 轴相切。
Example 3. Find the values of k for which x² + kx + 4 = 0 has two distinct real roots.
例 3. 求使方程 x² + kx + 4 = 0 有两个不同实根的 k 的取值范围。
Two distinct real roots require Δ > 0, so k² − 16 > 0, which gives k < −4 or k > 4.
有两个不同实根要求 Δ > 0，即 k² − 16 > 0，解得 k < −4 或 k > 4。
Example 4. Find the values of k for which x² + (k + 1)x + 9 = 0 has a repeated root.
例 4. 求使方程 x² + (k + 1)x + 9 = 0 有重根的 k 值。
−7. Always check whether the question asks for values (equality) or a range (inequality).
有重根要求 Δ = 0，即 (k + 1)² − 36 = 0，得 k + 1 = ±6，所以 k = 5 或 k = −7。务必看清题目要求的是具体取值（等式）还是取值范围（不等式）。
6. The Discriminant and Quadratic Inequalities | 判别式与二次不等式
A quadratic inequality such as ax² + bx + c > 0 for all real x is really a statement about the graph, and the discriminant is the fastest way to test it.
像“对一切实数 x 都有 ax² + bx + c > 0”这样的二次不等式，本质上是在描述图像的位置，而判别式是判断它的最快方法。
Because a parabola with a > 0 opens upwards, it stays strictly above the x-axis if and only if it never touches or crosses the axis. That happens exactly when Δ < 0.
因为 a > 0 的抛物线开口向上，它恒在 x 轴上方，当且仅当它从不与 x 轴相交或相切。这恰好对应 Δ < 0。
ax² + bx + c > 0 for all x ⟺ a > 0 and Δ < 0
ax² + bx + c < 0 for all x ⟺ a < 0 and Δ < 0
0 but the expression is negative everywhere, so it never satisfies the > 0 requirement.
注意两个条件缺一不可。取 a = 1，b = 0，c = −1，则 Δ = 4 > 0，于是 x² − 1 > 0 在 x = 0 时并不成立；又如 a = −1，b = 0，c = −1，此时 Δ = −4 < 0，但表达式处处为负，因此永远不满足 > 0 的要求。
For inequalities that are not required to hold for every x, the discriminant still helps: find the roots of the corresponding equation, sketch the parabola using the sign of a, and read the solution set off the sketch.
对于不要求对一切 x 都成立的不等式，判别式依然有用：先求出对应方程的根，再用 a 的符号画出抛物线草图，最后从图中直接读出解集。
7. Completing the Square and the Vertex | 配方法与顶点
The discriminant is closely tied to the completed-square form, which reveals the location of the vertex.
判别式与配方法的形式密切相关，而配方法能直接揭示顶点位置。
ax² + bx + c = a(x + b / (2a))² − Δ / (4a)
). If Δ < 0 then −Δ / (4a) > 0, confirming that the whole curve lies above the axis.
由于括号的平方恒不为负，当 a > 0 时表达式的最小值为 −Δ / (4a)，在 x = −b / (2a) 处取得。若 Δ < 0，则 −Δ / (4a) > 0，这正说明整条曲线位于 x 轴上方。
This gives a compact summary of the vertex: the x-coordinate is always −b / (2a), and the y-coordinate, which is the minimum or maximum value of the function, is −Δ / (4a).
由此可以简洁地总结顶点：横坐标恒为 −b / (2a)，纵坐标（即函数的最小值或最大值）为 −Δ / (4a)。
Exam questions often ask you to show that a quadratic is always positive. Either complete the square and show the minimum is positive, or state a > 0 and Δ < 0. Both are accepted; the second is faster.
考试中常要求证明某个二次式恒为正。你既可以配方并说明最小值大于零，也可以直接指出 a > 0 且 Δ < 0。两种写法都被接受，后者更快。
8. Intersections of a Line and a Curve | 直线与曲线的交点
One of the most common applications is finding how many times a straight line meets a quadratic curve.
最常见的应用之一，是判断一条直线与一条二次曲线相交多少次。
Substitute the equation of the line into the equation of the curve to eliminate y. The result is a quadratic equation in x, and its discriminant tells you the number of intersection points.
把直线方程代入曲线方程以消去 y，结果是一个关于 x 的二次方程，其判别式就告诉你交点的个数。
- Δ > 0: the line cuts the curve at two distinct points | 直线与曲线相交于两个不同点
- Δ = 0: the line is a tangent to the curve | 直线与曲线相切
- Δ < 0: the line does not meet the curve | 直线与曲线没有交点
m)x + 4 = 0. A tangent requires Δ = 0, so (4 − m)² − 16 = 0 and therefore m = 0 or m = 8.
例如，求使直线 y = mx + 3 与曲线 y = x² + 4x + 7 相切的 m 值。令 x² + 4x + 7 = mx + 3，得 x² + (4 − m)x + 4 = 0。相切要求 Δ = 0，即 (4 − m)² − 16 = 0，因此 m = 0 或 m = 8。
Remember that substituting a line into a curve sometimes produces a term in x² that cancels, leaving a linear equation. When that happens the line meets the curve exactly once for a reason unrelated to the discriminant, so check whether the x² coefficient is genuinely non-zero before using Δ.
请注意，把直线代入曲线后，x² 项有时会相互抵消，只剩一次方程。这时直线与曲线恰好有一个交点，但原因与判别式无关，所以在使用 Δ 之前应先确认 x² 的系数确实不为零。
9. Modelling Problems and Hidden Conditions | 建模问题与隐藏条件
In modelling questions the discriminant often appears disguised as a physical or geometrical condition.
在建模题中，判别式常常以物理或几何条件的形式隐藏出现。
hrases such as crosses at two points, two distinct solutions, or cuts the axis twice translate into Δ > 0. Phrases such as never reaches, misses the ground, or has no real solution translate into Δ < 0.
诸如“恰好接触”“擦过表面”“只有一个解”“与……相切”这类表述，都翻译成 Δ = 0；诸如“交于两点”“两个不同的解”“与坐标轴交于两处”翻译成 Δ > 0；而“从未到达”“错过地面”“没有实数解”则翻译成 Δ < 0。
In mechanics, a projectile following a parabolic path that must clear a wall of height h leads to an inequality on Δ or a direct comparison of the maximum value. In economics, a break-even condition often requires the revenue and cost curves to intersect, which again reduces to a discriminant condition.
在力学中，若抛体沿抛物线轨迹飞行且必须越过高度为 h 的墙，就会导出关于 Δ 的不等式，或直接比较函数的最大值。在经济学中，盈亏平衡条件往往要求收益曲线与成本曲线相交，这同样归结为判别式条件。
Always define your variable before writing Δ, and always state the inequality direction explicitly. A correct value of Δ with the wrong inequality sign earns no marks in the final step.
写 Δ 之前一定要先定义变量，并且明确写出不等号方向。Δ 的数值算对了但不等号写反，最后一步仍然拿不到分。
