Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程

📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation and graph sketching to real-life modelling. This revision guide explains the key methods, common pitfalls and exam-style strategies you need to solve quadratic equations confidently.

二次方程贯穿 IGCSE 数学大纲，从代数运算、图像绘制到实际建模。本复习指南讲解关键方法、常见错误和考试策略，帮助你自信地求解二次方程。

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 constants and a ≠ 0. The highest power of x is 2, which gives the equation its name.

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 为常数且 a ≠ 0。x 的最高次数为 2，因此得名。

If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential. The solutions of a quadratic equation are called roots, and they represent the x-values where the graph crosses the x-axis.

如果 a = 0，方程就变成一次方程，因此 a ≠ 0 的条件至关重要。二次方程的解称为根，它们表示图像与 x 轴交点的 x 值。

ax² + bx + c = 0, where a ≠ 0

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2. Solving by Factorising | 因式分解法解二次方程

When a quadratic expression factorises nicely, we can set each factor equal to zero and solve. For example, solve x² − 5x + 6 = 0 by writing (x − 2)(x − 3) = 0, giving x = 2 or x = 3.

当二次式容易因式分解时，我们可以令每个因式等于零来求解。例如，解 x² − 5x + 6 = 0，写成 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。

x² − 5x + 6 = 0 ⇒ (x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3

Always check by expanding the brackets. Watch for cases with a coefficient of x² greater than 1, such as 2x² + 5x − 3 = 0, which factorises to (2x − 1)(x + 3) = 0.

务必通过展开括号来检验。注意 x² 系数大于 1 的情况，例如 2x² + 5x − 3 = 0 可分解为 (2x − 1)(x + 3) = 0。

2x² + 5x − 3 = 0 ⇒ (2x − 1)(x + 3) = 0 ⇒ x = 1/2 or x = -3

Difference of squares is another common pattern, such as x² − 9 = (x − 3)(x + 3). Use it when there is no x term and both terms are perfect squares.

平方差是另一种常见模式，例如 x² − 9 = (x − 3)(x + 3)。当没有 x 项且两项都是完全平方时，可以使用这种方法。

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3. Solving by Completing the Square | 配方法解二次方程

Completing the square transforms a quadratic into the form a(x + p)² + q = 0. This method is especially useful when the quadratic does not factorise. For x² + 6x + 2 = 0, write (x + 3)² − 7 = 0, then solve.

配方法将二次方程转化为 a(x + p)² + q = 0 的形式。当二次式不能因式分解时，这种方法尤其有用。对于 x² + 6x + 2 = 0，写成 (x + 3)² − 7 = 0，然后求解。

x² + 6x + 2 = 0
x² + 6x = -2
x² + 6x + 9 = 7
(x + 3)² = 7
x = -3 ± √7

Remember to add the same value to both sides. Half of the coefficient of x is squared: (6/2)² = 9, and this value is added to complete the square.

记住要在等式两边同时加上相同的数。x 的系数一半的平方是 (6/2)² = 9，加上这个值即可完成配方。

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4. The Quadratic Formula | 二次求根公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is especially reliable when factorising is difficult or the roots are irrational.

二次求根公式可以解任何二次方程 ax² + bx + c = 0。当因式分解困难或根为无理数时，它特别可靠。

x = (-b ± √(b² − 4ac)) / (2a)

Substitute a, b and c carefully, including negative signs. For 3x² − 4x − 2 = 0, we have a = 3, b = -4 and c = -2.

代入 a、b、c 时要仔细，包括负号。对于 3x² − 4x − 2 = 0，a = 3，b = -4，c = -2。

x = (4 ± √(16 + 24)) / 6 = (4 ± √40) / 6 = (2 ± √10) / 3

Simplify the surd fully for the final answer. The formula is given in examinations, but you must still practise using it accurately and quickly.

最终答案要把根式完全化简。考试中会给出该公式，但你仍必须练习准确、快速地使用它。

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

The discriminant is the part of the quadratic formula under the square root: Δ = b² − 4ac. It tells us the nature of the roots without solving the equation fully.

判别式是二次求根公式中根号下的部分：Δ = b² − 4ac。它无需完整解方程即可告诉我们根的性质。

Δ = b² − 4ac

Discriminant condition	Nature of roots
Δ > 0	Two distinct real roots
Δ = 0	One repeated real root
Δ < 0	No real roots

If a question asks how many real solutions a quadratic has, calculate Δ first. For example, x² − 4x + 5 = 0 has Δ = 16 − 20 = -4, so it has no real roots.

如果题目问一个二次方程有多少个实数解，先计算 Δ。例如，x² − 4x + 5 = 0 的判别式 Δ = 16 − 20 = -4，因此没有实根。

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

The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upwards and has a U-shape; if a < 0, it opens downwards and has an inverted U-shape.

y = ax² + bx + c 的图像是抛物线。如果 a > 0，开口向上，呈 U 形；如果 a < 0，开口向下，呈倒 U 形。

The y-intercept is c, because when x = 0, y = c. The x-intercepts are the roots found by solving ax² + bx + c = 0. The vertex can be found by completing the square or by using x = -b/(2a).

y 轴截距为 c，因为当 x = 0 时，y = c。x 轴截距是通过解 ax² + bx + c = 0 得到的根。顶点可以通过配方法或使用 x = -b/(2a) 求得。

x = -b / (2a)

Knowing the shape, intercepts and vertex helps you sketch the graph accurately and solve related inequalities such as y > 0 or y < 0.

了解图像形状、截距和顶点有助于你准确绘制图像，并解决 y > 0 或 y < 0 等相关不等式。

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7. Using Graphs to Solve Equations | 利用图像解方程

You may be given a graph of y = x² + 2x − 3 and asked to solve x² + 2x − 3 = 5. Draw the horizontal line y = 5 on the same axes and read the x-coordinates of the intersection points.

考试可能给出 y = x² + 2x − 3 的图像，要求解 x² + 2x − 3 = 5。在同一坐标系中画出水平直线 y = 5，并读出交点的 x 坐标。

Those x-coordinates are the solutions of the equation. This method shows how algebraic solving and graphical interpretation are connected.

这些 x 坐标就是方程的解。这种方法体现了代数求解与图像解释之间的联系。

x² + 2x − 3 = 5 ⇔ x² + 2x − 8 = 0 ⇔ (x + 4)(x − 2) = 0

In an exam, if the graph is already drawn, use it rather than re-solving algebraically unless the question specifically asks for an algebraic method.

在考试中，如果图像已经画好，就直接使用图像，而不要重新用代数方法求解，除非题目明确要求使用代数方法。

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8. Applications and Word Problems | 实际应用与文字题

Word problems often require forming a quadratic equation from area, product, projectile motion or number relationships. Read carefully, define a variable and write an equation before solving.

文字题通常需要根据面积、乘积、抛体运动或数字关系建立二次方程。仔细读题，设出变量并写出方程，然后再求解。

Example: A rectangle has a length 3 cm longer than its width. If its area is 40 cm², find the width. Let the width be x cm, so the length is x + 3 cm. Then x(x + 3) = 40.

示例：一个矩形的长比宽多 3 cm。如果面积为 40 cm²，求宽。设宽为 x cm，则长为 x + 3 cm。于是 x(x + 3) = 40。

x(x + 3) = 40 ⇒ x² + 3x − 40 = 0 ⇒ (x + 8)(x − 5) = 0 ⇒ x = -8 or x = 5

Since a width cannot be negative, x = 5 cm is the only valid answer. Always check that your solution makes sense in the original context.

由于宽不能为负，x = 5 cm 是唯一合理的答案。务必检查解在原始情境中是否有意义。

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9. Common Mistakes and Exam Tips | 常见错误与考试技巧

Common mistakes include forgetting the negative sign when substituting b into the formula, dividing incorrectly when factorising, and ignoring the

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