Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程

📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程

Quadratic equations lie at the heart of the IGCSE Mathematics syllabus. They connect algebraic manipulation, graph sketching and problem solving, and they appear in both Paper 2 and Paper 4. This revision guide covers the standard form, three solution methods, the discriminant, graphs, inequalities and simultaneous equations.

二次方程是 IGCSE 数学课程的核心内容之一。它把代数运算、图像绘制和问题求解联系起来，在 Paper 2 和 Paper 4 中都会出现。本复习指南涵盖标准形式、三种解法、判别式、图像、不等式和联立方程。

1. Standard Form and Terminology | 标准形式与基本术语

and a ≠ 0. The condition a ≠ 0 is essential because if a = 0 the equation becomes linear.

一元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 为常数且 a ≠ 0。a ≠ 0 这一条件很重要，因为如果 a = 0，方程就变成了一次方程。

The terms have special names: ax² is the quadratic term, bx is the linear term and c is the constant term. You should always rearrange an equation into this form before choosing a solution method.

各项有专门名称：ax² 称为二次项，bx 称为一次项，c 称为常数项。在选择解法之前，应先把方程整理成标准形式。

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2. Solving by Factorising | 因式分解法

the left-hand side, then apply the zero product property: if AB = 0, then A = 0 or B = 0.

当二次方程有简单的整数根时，因式分解通常是最快的方法。先把方程右边化为 0，将左边因式分解，再使用零乘积性质：若 AB = 0，则 A = 0 或 B = 0。

6 = 0, write (x - 2)(x - 3) = 0. Therefore x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.

例如，解 x² - 5x + 6 = 0，可写成 (x - 2)(x - 3) = 0。因此 x - 2 = 0 或 x - 3 = 0，得到 x = 2 或 x = 3。

Always check your factors by expanding. A common mistake is writing (x + 2)(x - 3) for x² - 5x + 6; expanding gives x² - x - 6, which is not the original expression.

一定要通过展开来检验因式。常见错误是把 x² - 5x + 6 写成 (x + 2)(x - 3)；展开后得到 x² - x - 6，并不是原式。

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3. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This method is useful when a quadratic cannot be factorised easily, and it is also needed to find the vertex of a parabola.

配方法把 ax² + bx + c 写成 a(x + p)² + q 的形式。当二次方程不易因式分解时，这个方法很有用，而且求抛物线的顶点也需要它。

, which is 3, and write (x + 3)² - 9 + 1 = 0. This simplifies to (x + 3)² = 8, so x = -3 ± √8 = -3 ± 2√2.

对于 x² + 6x + 1 = 0，取 6 的一半，即 3，写成 (x + 3)² - 9 + 1 = 0。化简得 (x + 3)² = 8，所以 x = -3 ± √8 = -3 ± 2√2。

If the coefficient of x² is not 1, factor it out first. For 2x² + 8x + 3, write 2(x² + 4x) + 3, then complete the square inside the bracket.

如果 x² 的系数不是 1，要先把该系数提出来。例如 2x² + 8x + 3，写成 2(x² + 4x) + 3，再在括号内完成配方。

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4. The Quadratic Formula | 求根公式

The quadratic formula gives the solutions of ax² + bx + c = 0 directly:

x = (-b ± √b² - 4ac) / (2a)

求根公式可以直接给出 ax² + bx + c = 0 的解：

x = (-b ± √b² - 4ac) / (2a)

This formula works for every quadratic equation, including those with irrational or complex roots. In IGCSE, roots may be left in surd form unless a question asks for a decimal approximation.

这个公式适用于所有二次方程，包括根为无理数的情况。在 IGCSE 中，除非题目要求小数近似值，否则根可以保留为无理数形式。

and c carefully. For 2x² - 3x - 5 = 0, a = 2, b = -3, c = -5. Substituting gives x = (3 ± √9 + 40) / 4 = (3 ± 7) / 4, so x = 2.5 or x = -1.

使用公式前要仔细确定 a、b、c。对于 2x² - 3x - 5 = 0，a = 2，b = -3，c = -5。代入后得到 x = (3 ± √9 + 40) / 4 = (3 ± 7) / 4，所以 x = 2.5 或 x = -1。

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5. The Discriminant | 判别式

formula: Δ = b² - 4ac. It tells you the nature of the roots without solving the equation.

判别式是求根公式中平方根号下的表达式：Δ = b² - 4ac。它可以在不解方程的情况下判断根的性质。

f b² - 4ac = 0, there is one repeated real root. If b² - 4ac < 0, there are no real roots.

若 b² - 4ac > 0，方程有两个不相等的实根；若 b² - 4ac = 0，有一个重根；若 b² - 4ac < 0，没有实根。

2² - 4(1)(5) = 4 - 20 = -16, so it has no real roots. Its graph does not cross the x-axis.

例如，x² + 2x + 5 = 0 的判别式为 2² - 4(1)(5) = 4 - 20 = -16，所以它没有实根。它的图像不与 x 轴相交。

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6. Graphs of Quadratic Functions | 二次函数图像

. If a > 0 the parabola opens upwards and has a minimum point; if a < 0 it opens downwards and has a maximum point.

y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上，有最低点；若 a < 0，抛物线开口向下，有最高点。

2. The y-intercept is given by c, and the x-intercepts are the roots of ax² + bx + c = 0.

顶点可以通过配方法求得。对于 y = (x - 2)² + 3，顶点为 (2, 3)，对称轴为 x = 2。y 轴截距由 c 给出，x 轴截距是方程 ax² + bx + c = 0 的根。

Sketching a quadratic graph requires the vertex, the intercepts and the direction of opening. Always label these points clearly before drawing the smooth U-shaped curve.

绘制二次函数草图需要顶点、截距和开口方向。在画出光滑的 U 形曲线之前，应清楚标出这些点。

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7. Solving Quadratic Inequalities | 二次不等式

5x + 6 = 0, which are x = 2 and x = 3. These roots divide the number line into intervals.

解不等式 x² - 5x + 6 > 0 时，先求对应方程 x² - 5x + 6 = 0 的根，得到 x = 2 和 x = 3。这些根把数轴分成若干区间。

5 + 6 = -0.25 < 0, so it is excluded. For x > 3, choose x = 4, giving 16 - 20 + 6 = 2 > 0.

在每个区间取一个测试值：x < 2 时取 x = 0，得 0² - 5(0) + 6 = 6 > 0，所以该区间满足；2 < x < 3 时取 x = 2.5，得 6.25 - 12.5 + 6 = -0.25

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