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 are a central topic in IGCSE Mathematics, appearing in algebra, coordinate geometry, and real-world modelling. This article builds your skills from recognising the standard form to solving by factorisation, completing the square, and the quadratic formula, with exam-style tips throughout.

二次方程是 IGCSE 数学的核心主题，出现在代数、坐标几何和现实建模中。本文帮助你从识别标准形式到掌握因式分解法、配方法和二次公式，并贯穿考试技巧。

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

A quadratic equation is an equation in which the highest power of the unknown variable is 2. In IGCSE, the unknown is usually x, and the equation can often be rearranged into a form with x², x, and a constant term.

二次方程是未知数最高次数为 2 的方程。在 IGCSE 中，未知数通常为 x，方程通常可以整理为包含 x²、x 和常数项的形式。

equations. Equations such as x³ + x = 0 are not quadratic because the highest power is 3.

例如，x² − 5x + 6 = 0 和 2x² + 3x − 2 = 0 是二次方程。像 x³ + x = 0 这样的方程不是二次方程，因为最高次数为 3。

ax² + bx + c = 0

---

2. Standard Form and Coefficients | 标准形式与系数

e constants and a ≠ 0. The condition a ≠ 0 is essential: if a were zero, the x² term would disappear and the equation would be linear.

二次方程的标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。a ≠ 0 这一条件至关重要：若 a 为零，x² 项消失，方程就变成一次方程。

The coefficient a is called the leading coefficient. It controls the width and direction of the parabola when the quadratic expression is plotted. The constant c gives the y-intercept of the graph.

系数 a 称为首项系数。当二次表达式画成图像时，它控制抛物线的宽度和开口方向。常数 c 给出图像的 y 轴截距。

2x² − 3x + 1 = 0, a = 2, b = −3, c = 1

---

3. Solving by Factorisation | 因式分解法

Factorisation is usually the fastest method when the quadratic has integer roots. We rewrite ax² + bx + c as a product of two binomials, then apply the zero product property.

当二次方程有整数根时，因式分解通常是最快的方法。我们将 ax² + bx + c 改写为两个一次因式的乘积，然后应用零乘积性质。

umbers that multiply to 6 and add to −5. The numbers are −2 and −3, so (x − 2)(x − 3) = 0.

对于 x² − 5x + 6 = 0，我们寻找两个数，乘积为 6，和为 −5。这两个数是 −2 和 −3，因此 (x − 2)(x − 3) = 0。

x² − 5x + 6 = (x − 2)(x − 3)

---

4. The Zero Product Property | 零乘积性质

If the product of two factors is zero, then at least one factor must be zero. This property converts a factorised quadratic into two simple linear equations.

如果两个因式的乘积为零，那么至少有一个因式为零。这一性质将因式分解后的二次方程转化为两个简单的一次方程。

giving x = 2 or x = 3. Always write both solutions unless a question restricts the values.

由 (x − 2)(x − 3) = 0，我们令 x − 2 = 0 或 x − 3 = 0，得到 x = 2 或 x = 3。除非题目限制取值范围，否则要写出两个解。

(x − p)(x − q) = 0 ⇒ x = p or x = q

---

5. Completing the Square | 配方法

into a perfect square plus a constant. The key identity is x² + bx = (x + b/2)² − (b/2)².

当因式分解困难时，配方法将 x² + bx 转化为完全平方加上一个常数。核心恒等式为 x² + bx = (x + b/2)² − (b/2)²。

² − 9, so the equation becomes (x + 3)² − 4 = 0, giving (x + 3)² = 4 and x = −1 or x = −5.

对于 x² + 6x + 5 = 0，我们写 x² + 6x = (x + 3)² − 9，因此方程变为 (x + 3)² − 4 = 0，得到 (x + 3)² = 4，最终 x = −1 或 x = −5。

x² + bx = (x + b/2)² − (b/2)²

---

6. The Quadratic Formula | 二次公式

ion ax² + bx + c = 0. It is derived by completing the square on the general standard form.

二次公式可以解任意二次方程 ax² + bx + c = 0。它由一般标准形式配方推导而来。

x = (−b ± √b² − 4ac) ÷ (2a)

− 2 = 0, substitute a = 2, b = 3, c = −2. Then x = (−3 ± √(3² − 4×2×(−2))) ÷ (2×2) = (−3 ± √25) ÷ 4, so x = ½ or x = −2.

解 2x² + 3x − 2 = 0，代入 a = 2、b = 3、c = −2。则 x = (−3 ± √(3² − 4×2×(−2))) ÷ (2×2) = (−3 ± √25) ÷ 4，因此 x = ½ 或 x = −2。

---

广告 / Sponsored

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

quadratic formula. It tells us the number and type of roots without solving the equation.

判别式是二次公式平方根内的表达式 D = b² − 4ac。它告诉我们根的数量和类型，而无需实际解方程。

real root. If D < 0, there are no real roots, meaning the graph does not cross the x-axis.

若 D > 0，则有两个不同的实根；若 D = 0，则有一个重根；若 D < 0，则没有实根，即图像不与 x 轴相交。

D = b² − 4ac

---

8. Graphical Interpretation | 图像解释

The solutions of ax² + bx + c = 0 are the x-intercepts where the graph crosses the x-axis.

y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解是图像与 x 轴交点的横坐标。

If the discriminant is positive, the parabola crosses the x-axis twice. If it is zero, the vertex touches the x-axis. If it is negative, the parabola is entirely above or below the x-axis.

若判别式为正，抛物线与 x 轴相交两次；若判别式为零，顶点与 x 轴相切；若判别式为负，抛物线完全在 x 轴上方或下方。

y = ax² + bx + c

---

9. Word Problems and Applications | 应用题

IGCSE questions often embed quadratics in problems involving area, projectile motion, or number puzzles. Translate the words into an equation, then choose the most efficient solving method.

IGCSE 考题常将二次方程嵌入面积、抛体运动或数字谜题等情境。先将文字转化为方程，再选择最高效的解法。

40 = 0. Factorising gives x = 5 or x = −8; length must be positive, so the width is 5 cm.

例如，一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3，且 x(x + 3) = 40，即 x² + 3x − 40 = 0。因式分解得 x = 5 或 x = −8；长度必须为正，因此宽为 5 cm。

Always check whether solutions make sense in the original context. Negative solutions are often discarded for lengths, times, or counts.

始终检验解在原情境中是否有意义。负解通常不适用于长度、时间或数量。

---

10. Common Mistakes and Exam Tips | 常见错误与应试技巧

rise x² + 5x = 6 as x(x + 5) = 6 and then guess roots; rearrange to x² + 5x − 6 = 0 first.

一个常见错误是在因式分解前忘记先把方程整理为零。不能将 x² + 5x = 6 写为 x(x + 5) = 6 后猜测根；应先整理为 x² + 5x − 6 = 0。

rmula, especially when c is negative. Use brackets: b² − 4ac = (−3)² − 4(2)(−5) if needed.

另一个常见错误是二次公式中负号处理不当，尤其是 c

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）