Mastering Quadratics for IGCSE Mathematics | 掌握二次函数：IGCSE数学完全指南

📚 Mastering Quadratics for IGCSE Mathematics | 掌握二次函数：IGCSE数学完全指南

Quadratics form the backbone of IGCSE Mathematics, appearing in algebra, graphs, and problem-solving. This guide breaks down every essential skill you need, from expanding brackets to solving equations and sketching parabolas.

二次函数是IGCSE数学的核心内容，贯穿代数、图像与实际问题解决。本指南将系统梳理你所需掌握的全部关键技能，从展开括号、求解方程到绘制抛物线，一网打尽。

1. What Is a Quadratic Expression? | 什么是二次表达式？

A quadratic expression in one variable \(x\) is any expression that can be written in the form \(ax^2 + bx + c\), where \(a \neq 0\). The highest power of the variable is 2, which gives the expression its characteristic curved graph, called a parabola.

含有一个变量 \(x\) 的二次表达式是指可以写成 \(ax^2 + bx + c\) 的形式，其中 \(a \neq 0\)。变量的最高次数为2，这使得它的图像呈现出特有的弯曲形状，称为抛物线。

The coefficient \(a\) controls the width and direction of the parabola. If \(a > 0\), it opens upward; if \(a < 0\), it opens downward.
系数 \(a\) 控制抛物线的宽度与开口方向。若 \(a > 0\)，开口向上；若 \(a < 0\)，开口向下。
The constant term \(c\) is the y-intercept — the point where the graph crosses the y-axis.
常数项 \(c\) 是 y 轴截距，即图像与 y 轴相交的点。

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

Expanding means removing brackets by multiplying every term inside by every term outside. The most important rule is \((x + p)(x + q) = x^2 + (p+q)x + pq\).

展开意味着去掉括号，将括号外的每一项与括号内的每一项相乘。最重要的公式是 \((x + p)(x + q) = x^2 + (p+q)x + pq\)。

Factorising is the reverse process. For example, to factorise \(x^2 + 5x + 6\), find two numbers that multiply to 6 and add to 5 — those numbers are 2 and 3, so \(x^2 + 5x + 6 = (x+2)(x+3)\).

因式分解是相反的过程。例如，因式分解 \(x^2 + 5x + 6\)，需要找到两个数，乘积为6，和为5 —— 这两个数是2和3，因此 \(x^2 + 5x + 6 = (x+2)(x+3)\)。

3. Solving Quadratic Equations by Factorising | 用因式分解法解二次方程

A quadratic equation has the form \(ax^2 + bx + c = 0\). If the left side can be factorised, we use the zero product property: if \(AB = 0\), then \(A = 0\) or \(B = 0\).

二次方程具有形式 \(ax^2 + bx + c = 0\)。如果左侧能够因式分解，我们利用零乘积性质：若 \(AB = 0\)，则 \(A = 0\) 或 \(B = 0\)。

Solve \(x^2 – 7x + 10 = 0\)
Factor: \((x – 2)(x – 5) = 0\)
So \(x = 2\) or \(x = 5\)

Always rearrange the equation so that one side is zero before factorising. Do not divide by a variable unless you are sure it is never zero.

因式分解前务必先将方程整理为一边等于零。除非确定变量不为零，否则不要用变量去除等式两边。

4. The Quadratic Formula | 求根公式

Not every quadratic can be factorised easily. The quadratic formula solves any quadratic equation \(ax^2 + bx + c = 0\):

并非所有二次方程都能轻松因式分解。求根公式可以解任意二次方程 \(ax^2 + bx + c = 0\)：

\(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\)

Here, the value \(b^2 – 4ac\) is called the discriminant. It tells us how many real solutions exist:

这里的 \(b^2 – 4ac\) 称为判别式。它告诉我们实根的数量：

Discriminant	Number of real solutions	判别式	实根个数
\(b^2 – 4ac > 0\)	Two distinct real roots	\(b^2 – 4ac > 0\)	两个不同的实根
\(b^2 – 4ac = 0\)	One repeated real root	\(b^2 – 4ac = 0\)	一个重根
\(b^2 – 4ac < 0\)	No real roots	\(b^2 – 4ac < 0\)	无实根

Memorise this formula carefully. In the IGCSE exam, it is given on the formula sheet, but you must know how to substitute correctly and simplify the surd.

务必熟记此公式。在IGCSE考试中，公式表会提供它，但你必须会正确代入并化简根式。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic as \(a(x + h)^2 + k\). This elegant form reveals the vertex of the parabola immediately.

配方法将二次式改写为 \(a(x + h)^2 + k\)。这种简洁的形式立刻揭示抛物线的顶点。

\(x^2 + 6x + 5 = (x + 3)^2 – 9 + 5 = (x + 3)^2 – 4\)

The vertex is at \((-3, -4)\) because the expression inside the bracket is zero when \(x = -3\), and the constant term \(-4\) is the minimum value.

顶点在 \((-3, -4)\)，因为当 \(x = -3\) 时括号内为零，而常数项 \(-4\) 就是最小值。

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

当 \(x^2\) 的系数不为1时，先提取系数。例如，\(2x^2 – 8x + 3 = 2(x^2 – 4x) + 3 = 2[(x – 2)^2 – 4] + 3 = 2(x – 2)^2 – 5\)。

6. Solving Equations by Completing the Square | 用配方法解方程

Completing the square also gives a powerful way to solve quadratics, especially when factorisation fails. Let’s solve \(x^2 + 6x + 5 = 0\) using this method:

配方法也是解二次方程的有力工具，尤其是在因式分解行不通时。我们用配方法解 \(x^2 + 6x + 5 = 0\)：

\((x + 3)^2 – 4 = 0 \Rightarrow (x + 3)^2 = 4 \Rightarrow x + 3 = \pm 2 \Rightarrow x = -1 \text{ or } x = -5\)

This method works for every quadratic, and it directly connects to the vertex form of the graph.

这种方法适用于所有二次方程，并且直接与图像的顶点式相关联。

7. Sketching Graphs of Quadratics | 绘制二次函数图像

To sketch a parabola accurately, you need to find three key features: the roots (where it crosses the x-axis), the y-intercept, and the vertex (turning point).

要准确绘制抛物线，需要找到三个关键要素：根（与 x 轴的交点）、y 轴截距以及顶点（转向点）。

Roots: solve \(ax^2 + bx + c = 0\).
根：解方程 \(ax^2 + bx + c = 0\)。
y-intercept: set \(x = 0\), giving \(y = c\).
y 轴截距：令 \(x = 0\)，得到 \(y = c\)。
Vertex: use \(x = -\frac{b}{2a}\), then substitute to find y. Or use the completed square form.
顶点：使用 \(x = -\frac{b}{2a}\)，然后代回求 y；或者使用配方法后的顶点式。

Always check the sign of \(a\) to know which way the parabola opens. Plot the roots first, then the vertex, then draw a smooth symmetric curve.

始终检查 \(a\) 的符号以确定开口方向。先标出根，再标出顶点，最后画出平滑对称的曲线。

8. The Discriminant and Nature of Roots | 判别式与根的性质

The discriminant \(b^2 – 4ac\) does more than tell us how many roots exist; it also tells us whether the roots are rational or irrational.

判别式 \(b^2 – 4ac\) 不仅告诉我们根的数量，还告诉我们根是有理数还是无理数。

If the discriminant is a perfect square and greater than zero, the roots are rational and the quadratic can be factorised. If it is positive but not a perfect square, the roots are irrational surds.

如果判别式大于零且是完全平方数，根为有理数，二次式可以因式分解。如果判别式正但不是完全平方数，根为无理根式。

In IGCSE questions, you may be asked to find the range of values of \(k\) for which the equation has real roots. Set \(b^2 – 4ac \geq 0\) and solve the resulting inequality.

在IGCSE题目中，你可能会被要求找出使方程有实根的 \(k\) 的取值范围。令 \(b^2 – 4ac \geq 0\) 并解所得不等式即可。

9. Quadratic Inequalities | 二次不等式

Solving a quadratic inequality like \(x^2 – 5x + 6 < 0\) requires a graph-based understanding. First factorise: \((x - 2)(x - 3) < 0\). The roots are 2 and 3, and the parabola opens upward, so the expression is negative between the roots.

解二次不等式如 \(x^2 – 5x + 6 < 0\) 时需要基于图像理解。首先因式分解：\((x - 2)(x - 3) < 0\)。根为2和3，抛物线开口向上，因此表达式在两根之间为负。

Solution: \(2 < x < 3\)

For \(x^2 – 5x + 6 > 0\), the solution is \(x < 2\) or \(x > 3\). Always draw a quick sketch or use number lines to avoid sign mistakes.

对于 \(x^2 – 5x + 6 > 0\)，解为 \(x < 2\) 或 \(x > 3\)。务必画简图或使用数轴以避免符号错误。

10. Applications: Word Problems | 应用：文字应用题

Quadratics appear in real-world contexts such as projectile motion, area optimisation, and business profit models. A classic example is finding the dimensions of a rectangle given its area and perimeter.

二次函数出现在现实情境中，如抛体运动、面积优化和商业利润模型。一个经典问题是已知面积和周长，求矩形边长。

Suppose a rectangle has a perimeter of 20 m and an area of 24 m². Let the width be \(x\). Then the length is \(10 – x\). The area gives \(x(10 – x) = 24\), which simplifies to \(x^2 – 10x + 24 = 0\). Factorising gives \((x – 4)(x – 6) = 0\), so the sides are 4 m and 6 m.

设一个矩形周长为20米，面积为24平方米。设宽为 \(x\)，则长为 \(10 – x\)。面积方程为 \(x(10 – x) = 24\)，化简得 \(x^2 – 10x + 24 = 0\)。因式分解得 \((x – 4)(x – 6) = 0\)，因此边长分别为4米和6米。

11. Common Mistakes to Avoid | 常见错误避坑指南

Many students lose marks from small but avoidable errors. Here are the most frequent pitfalls we see in IGCSE quadratic questions.

许多学生因细小但可避免的错误而失分。以下是我们常见的IGCSE二次函数题目易错点。

Forgetting to set the equation equal to zero before factorising.
因式分解前忘记将方程化为零。
Misplacing the sign in the quadratic formula: always write \(-b\), not \(b\).
求根公式中符号错误：务必写 \(-b\)，不是 \(b\)。
Simplifying \(\sqrt{b^2 – 4ac}\) incorrectly — do not take the square root of each term separately.
错误化简 \(\sqrt{b^2 – 4ac}\) —— 不能分别对每一项开方。
When sketching, forgetting the y-intercept or plotting the vertex in the wrong direction.
画图时忘记 y 轴截距，或顶点方向标错。

12. Practice Strategy | 练习策略

To truly master quadratics, you must practise through spaced repetition. Begin with pure factorising drills, then progress to solving equations, then sketching, and finally word problems.

要真正掌握二次函数，你必须通过间隔重复来练习。先从纯因式分解训练开始，然后过渡到解方程、画图，最后是应用题。

Try to complete at least five mixed-practice questions daily. Mark your mistakes, categorise them (algebra, sign, method), and revisit the relevant section of this guide before the next day’s practice.

每天至少完成五道混合练习题。标记错误，将其分类（代数、符号、方法），并在第二天练习前重新阅读本指南的相关章节。

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Mastering Quadratics for IGCSE Mathematics | 掌握二次函数：IGCSE数学完全指南