📚 The Discriminant of a Quadratic Equation | 二次方程的判别式
In IB Mathematics, understanding the discriminant of a quadratic equation is crucial for analysing the nature of roots without fully solving the equation. The discriminant, often denoted by the Greek letter Δ (Delta), determines whether the roots are real and distinct, real and equal, or non-real complex conjugates. This concept not only connects algebra with graphing but also appears frequently in exam problems involving unknown coefficients and intersections of curves.
在IB数学中，理解二次方程的判别式对于不解方程而分析根的性质至关重要。判别式通常用希腊字母Δ（Delta）表示，它决定了根是相异实数、相等实数还是非实数的共轭复数。这个概念不仅将代数与图形联系起来，还经常出现在涉及未知系数和曲线交点的考题中。
1. Definition of the Discriminant | 判别式的定义
For a quadratic equation in standard form ax² + bx + c = 0 (with a ≠ 0), the discriminant is defined as:
Δ = b² − 4ac
This expression appears under the square root in the quadratic formula: x = [−b ± √(b² − 4ac)] / (2a).
对于标准形式 ax² + bx + c = 0（且 a ≠ 0）的二次方程，判别式定义为：
Δ = b² − 4ac
该表达式位于求根公式的根号下：x = [−b ± √(b² − 4ac)] / (2a)。
2. The Quadratic Formula and the Discriminant | 二次求根公式与判别式
The quadratic formula provides the solutions to any quadratic equation. Notice that the term √(b² − 4ac) is the discriminant. The ± sign indicates that there can be two roots, and their nature is entirely determined by the value of Δ. If Δ is positive, the square root yields a real number; if zero, the root simplifies to −b/(2a); if negative, the square root involves an imaginary number, leading to complex roots.
二次求根公式给出了任何二次方程的解。注意，√(b² − 4ac) 就是判别式的平方根。± 号表示可能有两个根，它们的性质完全由 Δ 的值决定。若 Δ 为正，平方根得到实数；若为零，根简化为 −b/(2a)；若为负，平方根包含虚数，从而产生复数根。
3. Three Cases Based on Δ | 基于 Δ 的三种情况
The discriminant classifies the roots into three distinct categories:
- Δ > 0: The equation has two distinct real roots.
- Δ = 0: The equation has exactly one real root (a repeated or double root).
- Δ < 0: The equation has no real roots; instead it has two complex conjugate roots.
判别式将根分为三种不同情况：
- Δ > 0：方程有两个不相等的实根。
- Δ = 0：方程恰好有一个实根（重根）。
- Δ < 0：方程没有实根，而有一对共轭复根。
4. Case 1: Δ > 0 – Two Distinct Real Roots | 情况一：Δ > 0 – 两个不相等的实根
When the discriminant is positive, the quadratic equation intersects the x-axis at two distinct points. If Δ is a perfect square, the roots are rational; otherwise, they are irrational surds. For example, x² − 5x + 6 = 0 gives Δ = 25 − 24 = 1, so roots are real, distinct, and rational (x = 2 and x = 3).
当判别式为正时，二次方程的图像与 x 轴交于两个不同的点。如果 Δ 是完全平方数，则根为有理数；否则为无理根式。例如 x² − 5x + 6 = 0，Δ = 25 − 24 = 1，因此根为不相等的有理实根（x = 2 和 x = 3）。
Roots: x = [−b ± √Δ] / (2a)
根：x = [−b ± √Δ] / (2a)
5. Case 2: Δ = 0 – Two Equal Real Roots | 情况二：Δ = 0 – 两个相等的实根
A discriminant of zero means the quadratic is a perfect square trinomial and can be factored as a(x − r)² = 0. The parabola touches the x-axis at exactly one point, the vertex. The single (repeated) root is given by x = −b/(2a). For instance, x² − 6x + 9 = 0 yields Δ = 36 − 36 = 0, giving the double root x = 3.
判别式为零意味着二次式是完全平方三项式，可分解为 a(x − r)² = 0。抛物线在顶点处恰好与 x 轴相切于一点。唯一（重）根由 x = −b/(2a) 给出。例如 x² − 6x + 9 = 0，Δ = 36 − 36 = 0，得到重根 x = 3。
6. Case 3: Δ < 0 – Complex Conjugate Roots | 情况三：Δ < 0 – 共轭复根
When the discriminant is negative, the quadratic has no real x-intercepts. However, in the field of complex numbers, the equation has two complex conjugate roots of the form p ± iq, where p = −b/(2a) and q = √(−Δ)/(2a). In IB Analysis & Approaches, complex numbers are studied; in Applications & Interpretation, the emphasis is often on stating that there are no real solutions, but the discriminant still informs the nature of the roots.
当判别式为负时，二次式没有实数 x 截距。然而在复数范围内，方程有两个共轭复数根，形式为 p ± iq，其中 p = −b/(2a)，q = √(−Δ)/(2a)。在 IB 数学分析与方法的课程中学习复数；在应用与解释课程中，通常强调没有实数解，但判别式依然指示了根的性质。
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