Quadratic Equations and Graphs for IGCSE Mathematics | IGCSE 数学：二次方程与图像

📚 Quadratic Equations and Graphs for IGCSE Mathematics | IGCSE 数学：二次方程与图像

Quadratic expressions appear throughout the IGCSE Mathematics syllabus, from basic algebra skills to graph sketching and real-world problem solving. This article provides a structured revision guide covering factorisation, completing the square, the quadratic formula, and the key features of parabolic graphs. Work through each section carefully and use the practice prompts to test your understanding before the final examination.

二次表达式贯穿 IGCSE 数学大纲的始终，从基础代数运算到图像绘制和实际问题求解都离不开它。本文提供一份系统的复习指南，涵盖因式分解、配方法、求根公式以及抛物线图像的主要特征。请仔细学习每一节，并在最终考试前使用练习提示来检测自己的掌握程度。

1. Understanding Quadratic Expressions | 认识二次表达式

A quadratic expression is any expression of the form ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which gives the expression its distinctive curved graph and its two possible solutions when set equal to zero.

二次表达式是形如 ax² + bx + c 的表达式，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数为 2，这使得它的图像具有独特的曲线形状，并且当表达式等于零时，方程最多有两个解。

For example, 3x² – 5x + 2, -x² + 4x and 2x² – 8 are all quadratic expressions. The coefficient a controls the width and direction of the parabola, while c gives the y-intercept when the expression is written as y = ax² + bx + c.

例如，3x² – 5x + 2、-x² + 4x 和 2x² – 8 都是二次表达式。系数 a 控制抛物线的开口方向和宽窄，而常数 c 给出表达式写成 y = ax² + bx + c 时图像在 y 轴上的截距。

If a > 0, the parabola opens upwards.
If a < 0, the parabola opens downwards.
If a 为正值，抛物线开口向上。
If a 为负值，抛物线开口向下。

2. Expanding Double Brackets | 展开双括号

Before solving quadratic equations, you must be able to expand products of two binomials accurately. The most reliable method is FOIL: multiply the First terms, Outer terms, Inner terms and Last terms, then collect like terms.

在解二次方程之前，你必须能够准确地展开两个二项式的乘积。最可靠的方法是 FOIL：分别将首项、外项、内项和末项相乘，然后合并同类项。

Expand (x + 3)(x – 2):

展开 (x + 3)(x – 2)：

(x + 3)(x – 2) = x² – 2x + 3x – 6 = x² + x – 6

When a coefficient is attached to one or both brackets, expand the brackets first and then multiply through by the coefficient. For example, 2(x + 1)(x – 4) = 2(x² – 3x – 4) = 2x² – 6x – 8.

当一个或两个括号前有系数时，应先展开括号，再将系数乘入各项。例如，2(x + 1)(x – 4) = 2(x² – 3x – 4) = 2x² – 6x – 8。

3. Factorising Quadratics with a = 1 | 首项系数为 1 的因式分解

Factorising is the reverse of expanding. To factorise x² + bx + c, look for two numbers that multiply to give c and add to give b. This is usually the fastest method for solving quadratic equations when the leading coefficient is 1.

因式分解是展开的逆运算。要对 x² + bx + c 进行因式分解，需要找到两个数，使它们的乘积等于 c，和等于 b。当首项系数为 1 时，这通常是解二次方程最快的方法。

Factorise x² + 7x + 10:

对 x² + 7x + 10 进行因式分解：

x² + 7x + 10 = (x + 2)(x + 5)

because 2 × 5 = 10 and 2 + 5 = 7. If c is negative, the two numbers will have opposite signs. If c is positive and b is negative, both numbers will be negative.

因为 2 × 5 = 10 且 2 + 5 = 7。如果 c 为负数，两个数异号；如果 c 为正数且 b 为负数，两个数都为负数。

4. Factorising Quadratics with a ≠ 1 | 首项系数不为 1 的因式分解

When a ≠ 1, factorising becomes slightly more demanding. One effective approach is the ac method: multiply a and c, find two numbers that multiply to ac and add to b, split the middle term, then factor by grouping.

当 a ≠ 1 时，因式分解会稍复杂一些。一种有效的方法是 ac 法：将 a 与 c 相乘，找到两个数，使它们的乘积为 ac 且和为 b，拆开中间项，再分组分解。

Factorise 2x² + 7x + 3:

对 2x² + 7x + 3 进行因式分解：

2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)

Check your answer by expanding: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. Always verify factorisation by expanding mentally or on paper.

通过展开检验答案：(2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3。因式分解后一定要在草稿上展开验证。

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

Once a quadratic equation is written in the form (px + q)(rx + s) = 0, we use the zero product property: if two factors multiply to zero, at least one factor must be zero. Set each bracket equal to zero and solve for x.

一旦二次方程写成 (px + q)(rx + s) = 0 的形式，就可以使用零乘积性质：如果两个因式的乘积为零，则至少有一个因式为零。令每个括号分别等于零，然后解出 x。

Solve x² – x – 12 = 0:

解方程 x² – x – 12 = 0：

(x – 4)(x + 3) = 0

x – 4 = 0 or x + 3 = 0

x = 4 or x = -3

Always present both solutions clearly. IGCSE mark schemes often award one method mark for correct factorisation and one accuracy mark for each correct root.

一定要清楚地写出两个解。IGCSE 评分标准通常会给正确的因式分解一个方法分，再给每一个正确的根一个准确分。

6. Completing the Square | 配方法

Completing the square transforms a quadratic into the form a(x + p)² + q. This form is extremely useful for finding the vertex of a parabola, solving equations, and proving that an expression is always positive or always negative.

配方法将二次式转化为 a(x + p)² + q 的形式。这种形式对于求抛物线的顶点、解方程以及证明一个表达式恒为正或恒为负都非常有用。

Rewrite x² + 6x + 5 in completed square form:

将 x² + 6x + 5 写成完全平方形式：

x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4

The vertex of the graph y = (x + 3)² – 4 is at (-3, -4). In general, y = a(x + p)² + q has its vertex at (-p, q).

图像 y = (x + 3)² – 4 的顶点坐标是 (-3, -4)。一般来说，y = a(x + p)² + q 的顶点坐标为 (-p, q)。

7. The Quadratic Formula and the Discriminant | 求根公式与判别式

When factorisation is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by the formula below. It is supplied in some IGCSE papers but must be memorised for non-calculator papers and applied accurately.

当因式分解困难或不可行时，应使用求根公式。对于 ax² + bx + c = 0，其解由下列公式给出。部分 IGCSE 试卷会提供该公式，但在非计算器试卷中必须牢记并准确运用。

x = (-b ± √(b² – 4ac)) / (2a)

The expression under the square root, Δ = b² – 4ac, is called the discriminant. It tells us how many real solutions exist without solving the equation fully.

平方根下的式子 Δ = b² – 4ac 称为判别式。它可以在不完整解方程的情况下告诉我们方程有多少个实数解。

If Δ > 0, there are two distinct real roots.
If Δ = 0, there is one repeated real root.
If Δ < 0, there are no real roots.
若 Δ > 0，方程有两个不同的实数根。
若 Δ = 0，方程有一个重根。
若 Δ < 0，方程没有实数根。

8. Graphs of Quadratic Functions | 二次函数的图像

The graph of y = ax² + bx + c is a parabola. Its most important features are the y-intercept, the x-intercepts, the axis of symmetry and the vertex. A sketch should show all of these clearly and be labelled with the relevant coordinates where possible.

y = ax² + bx + c 的图像是一条抛物线。最重要的特征包括 y 轴截距、x 轴截距、对称轴和顶点。草图应清晰地显示这些特征，并尽可能标出相关坐标。

Feature 特征	How to find it 如何求得
y-intercept	Set x = 0; y = c
x-intercepts	Set y = 0 and solve the quadratic
Axis of symmetry	x = -b / (2a)
Vertex	Substitute x = -b / (2a) into the equation

If the quadratic is written in completed square form, the vertex can be read directly. Otherwise, use the axis of symmetry to find it.

如果二次式写成完全平方形式，顶点可以直接读出；否则可利用对称轴求出顶点。

9. Solving Quadratic Equations Graphically | 用图像法解二次方程

Quadratic equations can also be solved by reading the x-intercepts from a graph. The solutions to ax² + bx + c = 0 are the values of x where the curve y = ax² + bx + c crosses the x-axis.

二次方程也可以通过读取图像上 x 轴截距来求解。方程 ax² + bx + c = 0 的解就是曲线 y = ax² + bx + c 与 x 轴相交处的 x 值。

If the curve crosses the x-axis at x = -1 and x = 3, then the solutions are x = -1 and x = 3. If the curve only touches the x-axis at one point, the equation has one repeated solution.

如果曲线在 x = -1 和 x = 3 处穿过 x 轴，则方程的解为 x = -1 和 x = 3。如果曲线只在一点接触 x 轴，则方程有一个重根。

Graphical methods are especially useful when solving two simultaneous equations, one linear and one quadratic. The points of intersection give the pairs of x and y values that satisfy both equations.

当求解一个一次方程与一个二次方程组成的方程组时，图像法尤其有用。两条曲线的交点坐标就是同时满足两个方程的 x 和 y 值。

10. Applications and Word Problems | 应用题与文字题

Many IGCSE questions embed quadratics in real-life contexts such as area, projectile motion, profit and physics. Translate the words into an equation, solve it, and then interpret the answer in the original context. Discard any solution that does not make practical sense, such as a negative length.

许多 IGCSE 题目将二次方程嵌入面积、抛体运动、利润和物理等实际情境中。先将文字转化为方程，求解后再将答案代回原情境进行解释。任何不符合实际意义的解，如负数长度，都应舍去。

Example: The area of a rectangular garden is 48 m². Its length is 2 m longer than its width. Find the dimensions.

例题：一个矩形花园的面积为 48 m²，其长比宽多 2 m。求矩形的长和宽。

Let width = x; length = x + 2; x(x + 2) = 48

x² + 2x – 48 = 0

(x + 8)(x – 6) = 0

x = -8 (reject) or x = 6

The width is 6 m and the length is 8 m. Always include units and a final sentence in word problems.

因此宽为 6 m，长为 8 m。在应用题中一定要写上单位并给出完整的结论句。

11. Common Mistakes and Exam Tips | 常见错误与考试建议

One common mistake is forgetting to set the equation to zero before factorising. You cannot solve x² + 3x = 10 by writing x(x + 3) = 10 and then setting each factor equal to 10. First rearrange to x² + 3x – 10 = 0, then factorise.

一个常见错误是在因式分解前忘记将方程整理为零的形式。你不能把 x² + 3x = 10 写成 x(x + 3) = 10，然后令每个因式等于 10。应先将方程移项为 x² + 3x – 10 = 0，再进行因式分解。

Another common mistake is losing a negative sign when using the quadratic formula. Substitute the values of a, b and c carefully, especially when c is negative. Use brackets around negative numbers to avoid sign errors.

另一个常见错误是在使用求根公式时遗漏负号。代入 a、b、c 的值时要特别小心，尤其是当 c 为负数时。建议将负数加上括号，以避免符号错误。

Always set the equation equal to zero before factorising.
Check factorisation by expanding.
State both roots unless the context gives a valid reason to discard one.
Label the vertex and intercepts when sketching graphs.
因式分解前一定要先让方程右边为零。
通过展开检验因式分解。
除非题目情境给出合理理由舍去一个根，否则两个解都要写出。
绘制图像时要标出顶点和截距。

12. Practice Prompts | 练习提示

Use these prompts to check your revision. Write out full working for each question and compare your methods with the approaches shown in this article.

使用以下提示来检测你的复习效果。请写出每道题的完整步骤，并将你的做法与本文所述方法进行比较。

Factorise and solve: x² – 8x + 15 = 0.
Solve using the quadratic formula: 2x² + 4x – 3 = 0.
Write y = x² – 4x + 1 in completed square form and state the vertex.
Sketch y = x² – 2x – 8, labelling the y-intercept, x-intercepts and vertex.
Find the value of k for which x² + kx + 9 = 0 has one repeated root.
对 x² – 8x + 15 = 0 进行因式分解并求解。
使用求根公式解方程：2x² + 4x – 3 = 0。
将 y = x² – 4x + 1 写成完全平方形式，并写出顶点坐标。
绘制 y = x² – 2x – 8 的图像，标出 y 轴截距、x 轴截距和顶点。
求使 x² + kx + 9 = 0 有一个重根的 k 值。

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Quadratic Equations and Graphs for IGCSE Mathematics | IGCSE 数学：二次方程与图像