The Discriminant | 判别式

📚 The Discriminant | 判别式

The discriminant is one of the shortest but most powerful tools in A-Level quadratics. It tells you immediately whether a quadratic equation ax² + bx + c = 0 has two distinct real roots, one repeated real root, or no real roots, without solving the equation. This idea extends to intersections of graphs, inequalities, and hidden conditions in modelling problems.

判别式是 A-Level 二次函数内容中简短却十分有力的工具。它能让你无需解方程，立刻判断二次方程 ax² + bx + c = 0 有两个不同实根、一个重根，还是没有实根。这一思想还可推广到图像交点、不等式以及建模问题中的隐藏条件。

1. What Is the Discriminant? | 什么是判别式

For any quadratic equation ax² + bx + c = 0, where a, b and c are real coefficients and a ≠ 0, the discriminant is the expression under the square root in the quadratic formula. It is usually denoted by the Greek letter Δ (delta).

对于任意二次方程 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，通常用希腊字母 Δ（delta）表示。

Δ = b² − 4ac

Because Δ depends only on the coefficients, you can determine the nature of the roots immediately after writing the equation in standard form. No factorising or solving is required.

由于 Δ 只取决于系数，你可以在将方程写成标准形式后立即判断根的性质，无需因式分解或解方程。

2. Derivation from the Quadratic Formula | 从求根公式推导

The quadratic formula states that the solutions of ax² + bx + c = 0 are:

求根公式表明 ax² + bx + c = 0 的解为：

x = (−b ± √(b² − 4ac)) / (2a)

The quantity b² − 4ac appears inside the square root. If the expression inside a square root is positive, you get a real number; if it is zero, you get zero; if it is negative, you do not get a real number

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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