Quadratic Equations & Functions | 二次方程与函数

📚 Quadratic Equations & Functions | 二次方程与函数

Quadratics form one of the most fundamental cornerstones of IGCSE Mathematics. From simple factorisation to graphing parabolas, understanding quadratic equations and functions is essential for success in both Paper 2 and Paper 4 of the examination. This article provides a complete walkthrough of the key concepts, techniques, and exam-style strategies you need to master this topic.

二次方程与函数是 IGCSE 数学中最核心的内容之一。无论是简单的因式分解，还是绘制抛物线图像，掌握二次方程与函数都是你在 Paper 2 和 Paper 4 中取得高分的关键。本文将为你系统梳理核心概念、解题技巧以及考试策略，帮助你在这一考点上全面突破。

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1. What is a Quadratic Equation? | 什么是二次方程

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 numbers and a ≠ 0. The reason we require a ≠ 0 is simple: if a = 0, the equation becomes linear and no longer involves an x² term.

二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为实数，且 a ≠ 0。要求 a ≠ 0 的原因很简单：如果 a = 0，方程就变成了一次方程，不再包含 x² 项。

A quadratic function takes the form y = ax² + bx + c. When drawn on a coordinate plane, its graph is always a smooth U-shaped or n-shaped curve called a parabola. The coefficient a determines the orientation: if a > 0 the parabola opens upward, while if a < 0 it opens downward.

二次函数的形式为 y = ax² + bx + c。在坐标平面上，它的图像总是一条平滑的 U 形或 ∩ 形曲线，称为抛物线。系数 a 决定开口方向：当 a > 0 时开口向上，当 a < 0 时开口向下。

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2. Expanding and Factorising | 展开与因式分解

Expanding is the process of removing brackets. The key identity is (x + p)(x + q) = x² + (p + q)x + pq. For example, (x + 3)(x − 5) = x² − 2x − 15. Notice that the coefficient of x is obtained by adding 3 and −5, while the constant term is obtained by multiplying them.

展开是去括号的过程。核心恒等式为 (x + p)(x + q) = x² + (p + q)x + pq。例如，(x + 3)(x − 5) = x² − 2x − 15。注意 x 的系数由 3 与 −5 相加得到，而常数项由它们相乘得到。

Factorisation is the reverse process. To factorise x² + bx + c, we search for two numbers whose product is c and whose sum is b. The difference of two squares formula a² − b² = (a + b)(a − b) is also frequently tested, such as x² − 9 = (x + 3)(x − 3).

因式分解是展开的逆过程。要分解 x² + bx + c，我们需要找到两个数，使它们的乘积为 c、和为 b。平方差公式 a² − b² = (a + b)(a − b) 也是常考内容，例如 x² − 9 = (x + 3)(x − 3)。

Expression	Factorised Form
x² + 7x + 12	(x + 3)(x + 4)
x² − 5x − 14	(x − 7)(x + 2)
4x² − 25	(2x + 5)(2x − 5)
x² + 6x + 9	(x + 3)²

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3. Solving by Factorisation | 用因式分解求解方程

The zero product property states that if AB = 0, then A = 0 or B = 0. This is the engine behind solving quadratics by factorisation. Once a quadratic is factorised as (x − p)(x − q) = 0, the solutions are simply x = p and x = q.

零积性质指出：如果 AB = 0，那么 A = 0 或 B = 0。这是用因式分解法求解二次方程的核心原理。当方程被分解为 (x − p)(x − q) = 0 后，解即为 x = p 和 x = q。

Worked example: Solve x² + 5x + 6 = 0. We need two numbers that multiply to 6 and add to 5; those numbers are 2 and 3. Hence (x + 2)(x + 3) = 0, giving x = −2 or x = −3.

例题：解 x² + 5x + 6 = 0。我们需要找乘积为 6、和为 5 的两个数，即 2 和 3。因此 (x + 2)(x + 3) = 0，解得 x = −2 或 x = −3。

Always rearrange the equation to the form ax² + bx + c = 0 before factorising. A common mistake is to try to factorise without moving all terms to one side first, which leads to invalid roots.

在因式分解之前，一定要先将方程整理为 ax² + bx + c = 0 的形式。一个常见错误是未将所有项移到等号一侧就急着分解，这样会得到无效的根。

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4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. For x² + bx, add and subtract (b/2)² to obtain x² + bx = (x + b/2)² − (b/2)². This technique is invaluable for finding the vertex of a parabola and for solving equations when factorisation is not straightforward.

配方法将二次式改写为 (x + p)² + q 的形式。对于 x² + bx，加上再减去 (b/2)²，可得 x² + bx = (x + b/2)² − (b/2)²。这一技巧在求抛物线顶点以及因式分解不便时解方程都非常实用。

x² + bx = (x + b/2)² − (b/2)²

Example: x² − 6x + 7 = (x − 3)² − 9 + 7 = (x − 3)² − 2. The vertex of y = x² − 6x + 7 is therefore (3, −2).

例如：x² − 6x + 7 = (x − 3)² − 9 + 7 = (x − 3)² − 2。因此抛物线 y = x² − 6x + 7 的顶点坐标为 (3, −2)。

When the coefficient of x² is not 1, factor it out first. For example: 2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3.

当 x² 的系数不为 1 时，先将其提出来。例如：2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3。

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5. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0 directly. It is derived by completing the square on the general form of the equation, and it works for every quadratic regardless of whether the roots are rational or irrational.

二次公式可以直接求解任意二次方程 ax² + bx + c = 0。它通过对一般形式进行配方推导得出，对任何二次方程均适用，无论根是有理数还是无理数。

x = (−b ± √(b² − 4ac)) ÷ 2a

To use the formula, identify a, b and c, substitute them carefully, and simplify. Watch your signs! For example, solving 2x² − 3x − 5 = 0 gives a = 2, b = −3, c = −5. Substituting yields x = (3 ± √49) ÷ 4 = (3 ± 7) ÷ 4, so x = 2.5 or x = −1.

使用公式时，先准确识别 a、b、c，代入后仔细化简，务必注意符号！例如，解 2x² − 3x − 5 = 0 时，a = 2，b = −3，c = −5。代入得 x = (3 ± √49) ÷ 4 = (3 ± 7) ÷ 4，所以 x = 2.5 或 x = −1。

The formula is especially useful when factorisation is difficult or impossible — for instance when the roots contain surds, such as x² + 2x − 3 = 0 producing roots with √3.

当因式分解困难甚至无法进行时，公式法尤其有用。例如，当根中含有无理数时，如 x² + 2x − 3 = 0 的根含有 √3。

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6. The Discriminant | 判别式

The expression b² − 4ac under the square root in the quadratic formula is called the discriminant, often denoted by the symbol Δ. It tells us the nature of the roots without actually solving the equation.

二次公式根号内的表达式 b² − 4ac 称为判别式，通常用符号 Δ 表示。它无需解方程就能告诉我们根的性质。

Discriminant	Nature of Roots	Graph
Δ > 0	Two distinct real roots	Crosses x-axis at two points
Δ = 0	One repeated real root	Touches x-axis at one point
Δ < 0	No real roots	Does not meet x-axis

If Δ > 0, the equation has two distinct real roots. If Δ = 0, it has exactly one repeated real root. If Δ < 0, there are no real roots, which means the parabola does not intersect the x-axis at all.

当 Δ > 0 时，方程有两个不同的实数根；当 Δ = 0 时，方程有一个重根；当 Δ < 0 时，方程没有实数根，即抛物线完全不与 x 轴相交。

Example: For x² + 4x + k = 0 to have two distinct real roots, we require Δ = 16 − 4k > 0, so k < 4. If k = 4, the equation has exactly one root; if k > 4, it has no real roots.

例如：要使 x² + 4x + k = 0 有两个不同的实数根，需要 Δ = 16 − 4

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