Solving Quadratic Equations: Methods & Applications | 二次方程的解法与应用

📚 Solving Quadratic Equations: Methods & Applications | 二次方程的解法与应用

A quadratic equation is one of the most fundamental topics in mathematics, appearing in algebra, geometry, physics and beyond. This revision guide covers every standard method of solution, the discriminant, Vieta’s formulas, graphical interpretation, and realistic applications, with exam-focused examples throughout.

二次方程是数学中最基础的主题之一，广泛出现在代数、几何、物理等领域。本复习指南涵盖所有标准解法、判别式、韦达定理、图象解释以及实际应用，并配有紧扣考试的例题。

1. Standard Form and Fundamentals | 标准形式与基本概念

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are real constants and a ≠ 0. The term “quadratic” comes from the Latin “quadratus”, meaning square, because the highest power of the variable x is 2.

二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是实数常数，且 a ≠ 0。”quadratic” 一词源自拉丁语 “quadratus”，意为”平方”，因为变量 x 的最高次数为 2。

Before solving any quadratic, always identify the coefficients a, b and c. For example, in 2x² − 3x + 1 = 0, we have a = 2, b = −3 and c = 1. If a term is missing, its coefficient is zero — for instance x² − 4 = 0 has b = 0.

在解任何二次方程之前，必须先确认系数 a、b 和 c。例如，在 2x² − 3x + 1 = 0 中，a = 2，b = −3，c = 1。若某项缺失，则其系数为零——例如 x² − 4 = 0 中 b = 0。

• A quadratic equation always has degree 2; the graph of y = ax² + bx + c is a parabola.

  二次方程的次数恒为 2；y = ax² + bx + c 的图象是一条抛物线。

• a ≠ 0 is essential; if a = 0 the equation becomes linear (bx + c = 0).

  a ≠ 0 至关重要；若 a = 0，方程退化为一次方程（bx + c = 0）。

• Every quadratic equation has exactly two roots (counting multiplicity) over the complex number system, but it may have fewer real roots.

  在复数范围内，每个二次方程恰好有两个根（按重数计），但实根的个数可能少于两个。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has rational factors. The method relies on the zero product property: if A × B = 0, then A = 0 or B = 0.

因式分解是当二次式含有有理因式时最快捷的方法。该方法依据零乘积性质：若 A × B = 0，则 A = 0 或 B = 0。

For a monic quadratic x² + bx + c, find two numbers whose product is c and whose sum is b. For ax² + bx + c with a ≠ 1, find two numbers whose product is a×c and whose sum is b, then split the middle term and factor by grouping.

对于首项系数为 1 的二次式 x² + bx + c，找出两个数，使其积为 c、和为 b。对于 a ≠ 1 的二次式 ax² + bx + c，

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