📚 IB Mathematics: The Discriminant of Quadratic Equations Fully Explained | IB数学：二次方程判别式全解析
The discriminant is one of the most powerful shortcuts in the IB Mathematics syllabus. It allows us to determine the number and type of roots of a quadratic equation without actually solving the equation.
判别式是IB数学考纲中最强大的捷径之一。它让我们无需真正解方程，就能够判断一个二次方程有多少个根以及这些根的类型。
1. What Is the Discriminant? | 什么是判别式？
For a quadratic equation written in standard form, where a, b and c are real numbers and a is not zero:
对于一个写成标准型的二次方程，其中 a、b、c 为实数，且 a 不等于 0：
ax² + bx + c = 0
The discriminant is defined as the expression under the square-root sign in the quadratic formula:
判别式定义为二次求根公式中根号下的表达式：
Δ = b² − 4ac
The Greek letter Δ, read as "delta", is the standard symbol for the discriminant. It is also sometimes written as D in older textbooks.
希腊字母 Δ，读作“delta”，是判别式的标准符号。在部分较旧的教材中，它也会写作 D。
| Δ value | Number of real roots | Type of roots |
| Δ > 0 | 2 | Two distinct real roots |
| Δ = 0 | 1 | Two equal real roots, also called a repeated root |
| Δ < 0 | 0 | No real roots; two complex conjugate roots |
2. Where Does the Discriminant Come From? | 判别式是从哪里来的？
ic formula is derived by completing the square. Starting from ax² + bx + c = 0, we obtain:
二次求根公式是通过配方推导得到的。从 ax² + bx + c = 0 出发，可以得到：
x = (−b ± √Δ) / (2a), where Δ = b² − 4ac
are root of Δ is a positive real number, so the ± sign produces two different values of x.
如果 Δ 为正数，那么 Δ 的平方根就是一个正实数，正负号会给出两个不同的 x 值。
iving exactly one value. This is why we describe a zero discriminant as producing two equal roots.
如果 Δ 等于 0，平方根为 0，表达式就简化为 x = −b / (2a)，只能得到一个值。这就是为什么我们说判别式为 0 时产生两个相等的根。
If Δ is negative, the square root of a negative number is not real. The equation has no real solutions, but it does have two complex conjugate solutions involving i.
如果 Δ 为负数，负数的平方根在实数范围内不存在。方程没有实数解，但会有两个包含 i 的共轭复数解。
3. The Three Cases | 三种情形
Case 1: Δ = b² − 4ac > 0
情形一：Δ = b² − 4ac > 0
When the discriminant is positive, the quadratic equation has two distinct real roots. The parabola crosses the x-axis at two different points.
当判别式为正时，二次方程有两个不相等的实数根。抛物线会在两个不同的点穿过 x 轴。
Case 2: Δ = 0
情形二：Δ = 0
When the discriminant is zero, the equation has two equal real roots, also called a double root. The parabola touches the x-axis at exactly one point, and that point is the vertex.
当判别式为 0 时，方程有两个相等的实数根，也称为重根。抛物线只在一点处与 x 轴相切，这个点就是顶点。
Case 3: Δ < 0
情形三：Δ < 0
When the discriminant is negative, the equation has no real roots. The parabola does not intersect or touch the x-axis at all. It lies either entirely above the x-axis or entirely below it.
当判别式为负时，方程没有实数根。抛物线不与 x 轴相交也不相切。它要么完全位于 x 轴上方，要么完全位于 x 轴下方。
4. Worked Examples | 完整例题分析
Example 1: x² − 5x + 6 = 0
例 1：x² − 5x + 6 = 0
Here a = 1, b = −5 and c = 6.
这里 a = 1，b = −5，c = 6。
Δ = (−5)² − 4 × 1 × 6 = 25 − 24 = 1
Since Δ = 1 > 0, there are two distinct real roots. Solving gives x = 2 or x = 3.
因为 Δ = 1 > 0，所以有两个不相等的实数根。解得 x = 2 或 x = 3。
Example 2: x² − 4x + 4 = 0
例 2：x² − 4x + 4 = 0
Here a = 1, b = −4 and c = 4.
这里 a = 1，b = −4，c = 4。
Δ = (−4)² − 4 × 1 × 4 = 16 − 16 = 0
Since Δ = 0, the equation has a repeated real root. The double root is x = 2.
因为 Δ = 0，方程有一个重根。这个重根为 x = 2。
Example 3: x² + 2x + 5 = 0
例 3：x² + 2x + 5 = 0
Here a = 1, b = 2 and c = 5.
这里 a = 1，b = 2，c = 5。
Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16
Since Δ < 0, there are no real roots. The solutions are complex: x = −1 ± 2i.
因为 Δ < 0，所以没有实数根。解为复数：x = −1 ± 2i。
5. Graphical Interpretation | 图形意义
discriminant is deeply connected to the graph of the quadratic function y = ax² + bx + c.
判别式与二次函数 y = ax² + bx + c 的图像密切相关。
If Δ > 0, the graph cuts the x-axis in two distinct places.
如果 Δ > 0，图像与 x 轴有两个不同的交点。
If Δ = 0, the vertex lies exactly on the x-axis, so the graph is tangent to the x-axis.
如果 Δ = 0，顶点正好在 x 轴上，因此图像与 x 轴相切。
If Δ < 0 and a > 0, the graph lies completely above the x-axis.
如果 Δ < 0 且 a > 0，图像完全位于 x 轴上方。
If Δ < 0 and a < 0, the graph lies completely below the x-axis.
如果 Δ < 0 且 a < 0，图像完全位于 x 轴下方。
6. Discriminant and Tangency | 判别式与相切问题
One of the most common IB applications is finding the conditions under which a line is tangent to a parabola.
IB考试中最常见的应用之一，就是判断一条直线与一条抛物线何时相切。
intersects the parabola y = ax² + bx + c, we set the two expressions equal to each other:
要判断直线 y = mx + k 是否与抛物线 y = ax² + bx + c 相交，我们将两个表达式的 y 值相等：
ax² + bx + c = mx + k
Rearranging gives a new quadratic equation in x. The discriminant of this new quadratic tells us how many intersection points exist.
整理后得到一个关于 x 的新的二次方程。这个新二次方程的判别式决定了交点的个数。
Example: The line y = 2x + k is tangent to y = x² − x + 3. Find k.
例：直线 y = 2x + k 与 y = x² − x + 3 相切，求 k。
Set x² − x + 3 = 2x + k, which gives x² − 3x + 3 − k = 0.
令 x² − x + 3 = 2x + k，得到 x² − 3x + 3 − k = 0。
Δ = (−3)² − 4(3 − k) = 9 − 12 + 4k = 4k − 3
For tangency, we need Δ = 0, so 4k − 3 = 0 and therefore k = ¾.
相切要求 Δ = 0，因此 4k − 3 = 0，所以 k = ¾。
7. Using Inequalities to Find Unknown Parameters | 利用判别式求未知参数
Many IB questions ask for the range of values of an unknown parameter, such as k or p, that makes the equation satisfy a certain condition.
许多IB题目会要求我们求某个未知参数（如 k 或 p）的取值范围，使得方程满足特定条件。
Example: Find the set of values of k for which x² + 2kx + k = 0 has no real roots.
例：求 k 的取值范围，使得 x² + 2kx + k = 0 没有实数根。
Here a = 1, b = 2k and c = k.
这里 a = 1，b = 2k，c = k。
Δ = (2k)² − 4k = 4k² − 4k = 4k(k − 1)
For no real roots, we need Δ < 0:
若要没有实数根，需要 Δ < 0：
4k(k − 1) < 0
Solving this quadratic inequality gives 0 < k < 1.
解这个二次不等式，得到 0 < k < 1。
8. Discriminant and the Sign of a Quadratic Function | 判别式与二次函数的正负性
The discriminant also helps determine whether a quadratic expression is always positive or always negative.
判别式还能帮助我们判断一个二次表达式是否恒为正或恒为负。
If a > 0 and Δ < 0, then ax² + bx + c > 0 for all real values of x.
如果 a > 0 且 Δ < 0，那么对于所有实数 x，ax² + bx + c > 0 恒成立。
If a < 0 and Δ < 0, then ax² + bx + c < 0 for all real values of x.
如果 a < 0 且 Δ < 0，那么对于所有实数 x，ax² + bx + c < 0 恒成立。
Example: Show that x² + 4x + 7 is always positive.
例：证明 x² + 4x + 7 恒为正。
Δ = 4² − 4 × 1 × 7 = 16 − 28 = −12
Since a = 1 > 0 and Δ = −12 < 0, the expression is positive for every real value of x.
因为 a = 1 > 0 且 Δ = −12 < 0，该表达式对任意实数 x 都为正。
9. Common IB Exam Traps | 常见IB考试陷阱
pression is not a quadratic equation and the discriminant is not defined in the usual way.
忘记方程必须是二次方程。如果 a = 0，那么该表达式不是二次方程，判别式也不能按照通常意义讨论。
Confusing "two equal real roots" with "no roots". A double root is still a real root.
混淆“两个相等的实数根”和“没有根”。重根仍然是实数根。
Making sign errors when b is negative. Always substitute the whole value of b, including its sign.
当 b 为负数时出现符号错误。始终将 b 的整个值，包括符号，代入公式。
Writing "no roots" when the correct statement is "no real roots". In IB Higher Level questions, complex roots may be relevant.
写成“没有根”而不写“没有实数根”。在IB高级水平题目中，复数根也可能是考查内容。
Forgetting to check the leading coefficient when solving parameter questions. Sometimes the coefficient a itself depends on the parameter.
在求解参数问题时忘记检查二次项系数。有时系数 a 本身也依赖于该参数。
10. Practice Problem | 练习题目
Question: Find the values of k for which kx² + 2x + 1 = 0 has two distinct real roots.
问题：求 k 的取值范围，使得 kx² + 2x + 1 = 0 有两个不相等的实数根。
We require a = k ≠ 0 and Δ > 0.
我们需要 a = k ≠ 0，并且 Δ > 0。
Δ = 2² − 4 × k × 1 = 4 − 4k
For Δ > 0:
要满足 Δ > 0：
4 − 4k > 0 → k < 1
Combining with k ≠ 0 gives the final answer:
结合 k ≠ 0，得到最终答案：
k < 1 and k ≠ 0
were allowed, the equation would become linear and could not have two distinct real roots.
如果允许 k = 0，方程将变成一次方程，不可能有两个不相等的实数根。
11. Summary | 要点总结
The discriminant is Δ = b² − 4ac for ax² + bx + c = 0.
对于 ax² + bx + c = 0，判别式为 Δ = b² − 4ac。
Δ > 0 means two distinct real roots.
Δ > 0 表示两个不相等的实数根。
Δ = 0 means two equal real roots, or one repeated root.
Δ = 0 表示两个相等的实数根，也就是一个重根。
Δ < 0 means no real roots, but two complex conjugate roots.
Δ < 0 表示没有实数根，但有两个共轭复数根。
The discriminant is also used to study tangency, sign of a quadratic, and intersection problems.
判别式还可用于研究相切、二次函数的正负性以及交点问题。
Once you understand the discriminant, many IB questions become faster and less error-prone. Always write down the equation in standard form first, then compare coefficients carefully.
一旦理解了判别式，很多IB题目都会变得更快、更不容易出错。一定要先把方程写成标准形式，再仔细比较系数。
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