Mastering Quadratic Equations | 二次方程完全指南

📚 Mastering Quadratic Equations | 二次方程完全指南

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, geometry, and real-world problems. This guide is designed for teachers and students to master every essential skill step by step.

二次方程是 IGCSE 数学中最重要的话题之一。它们出现在代数、图像、几何和实际问题中。本指南面向教师和学生，帮助逐步掌握每一个核心技能。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which gives the equation its name.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次幂为 2，因此得名二次。

For example, 2x² – 5x + 3 = 0 is a quadratic equation. In this case, a = 2, b = -5, c = 3.

例如，2x² – 5x + 3 = 0 是一个二次方程。此时 a = 2，b = -5，c = 3。

If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential. Understanding the standard form helps you identify the coefficients and choose the best solution method.

如果 a = 0，方程就变成一次方程，所以 a ≠ 0 这一条件至关重要。理解标准形式有助于你识别系数并选择最佳解法。

2. Solving by Factorising | 因式分解法

Factorising is often the fastest method when the quadratic has simple integer roots. The idea is to rewrite ax² + bx + c as a product of two linear factors.

当二次方程具有简单的整数根时，因式分解通常是最快的方法。其思想是将 ax² + bx + c 改写为两个一次因式的乘积。

x² + 5x + 6 = (x + 2)(x + 3) = 0

Then, using the zero product property, x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.

然后利用零乘积性质，x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。

For equations with a leading coefficient greater than 1, such as 2x² + 5x + 2 = 0, you may need to split the middle term or use trial and error.

对于首项系数大于 1 的方程，如 2x² + 5x + 2 = 0，你可能需要拆项或尝试配凑因式。

2x² + 5x + 2 = (2x + 1)(x + 2) = 0

Here x = -½ or x = -2. Always expand your factors to check your work.

这里 x = -½ 或 x = -2。务必展开因式来检验计算。

3. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that works for any quadratic equation. It rewrites the equation in the form a(x – h)² + k = 0, which also reveals the turning point of the graph.

配方法是一种适用于任何二次方程的强大技巧。它将方程改写为 a(x – h)² + k = 0 的形式，并且能揭示图像的顶点。

Start with x² + 6x – 7 = 0. Take half of 6, square it to get 9, then add and subtract 9:

从 x² + 6x – 7 = 0 开始。取 6 的一半为 3，平方得 9，然后加减 9：

x² + 6x + 9 – 9 – 7 = 0 → (x + 3)² – 16 = 0

Now solve: (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = -7.

接着解： (x + 3)² = 16，所以 x + 3 = ±4，得 x = 1 或 x = -7。

When the coefficient of x² is not 1, first factor it out from the quadratic terms. For example, 2x² – 8x + 5 = 0 becomes 2(x² – 4x) + 5 = 0, then complete the square inside the bracket.

当 x² 的系数不是 1 时，先将其从二次项和一次项中提出。例如，2x² – 8x + 5 = 0 变为 2(x² – 4x) + 5 = 0，然后在括号内配方。

4. The Quadratic Formula | 二次公式

The quadratic formula provides a direct method for solving any quadratic equation. For ax² + bx + c = 0, the solutions are given by:

二次公式提供了一种直接求解任何二次方程的方法。对于 ax² + bx + c = 0，解为：

x = (-b ± √(b² – 4ac)) / 2a

This formula is derived from completing the square. It is especially useful when the equation cannot be factorised easily.

该公式由配方法推导得出。当方程不易因式分解时尤其有用。

For example, solve 3x² + 2x – 8 = 0. Here a = 3, b = 2, c = -8. Substitute:

例如，解 3x² + 2x – 8 = 0。这里 a = 3，b = 2，c = -8。代入：

x = (-2 ± √(4 – 4×3×(-8))) / 6 = (-2 ± √100) / 6 = (-2 ± 10) / 6

So x = 8/6 = 4/3 or x = -12/6 = -2. Always simplify surds and fractions.

因此 x = 8/6 = 4/3 或 x = -12/6 = -2。务必化简根式与分数。

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

The expression under the square root in the quadratic formula is called the discriminant, denoted by Δ = b² – 4ac. Its sign tells us how many real roots the equation has.

二次公式中根号内的表达式称为判别式，记作 Δ = b² – 4ac。它的符号告诉我们方程有多少个实数根。

Δ > 0: two distinct real roots
Δ > 0：两个不同的实数根
Δ = 0: one repeated real root
Δ = 0：一个重根（两个相等的实数根）
Δ < 0: no real roots (two complex roots)
Δ < 0：没有实数根（两个复数根）

For example, x² – 4x + 4 = 0 has Δ = 16 – 16 = 0, so it has exactly one solution x = 2.

例如，x² – 4x + 4 = 0 的 Δ = 16 – 16 = 0，因此它只有一个解 x = 2。

In IGCSE, you are often asked to determine whether a quadratic has real roots without solving it. This is a fast and reliable test.

在 IGCSE 考试中，你常被要求不解方程就判断二次方程是否有实数根。判别式是一种快速而可靠的检验方法。

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

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.

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

The x-intercepts of the graph are the real roots of the equation ax² + bx + c = 0. The y-intercept is c.

图像与 x 轴的交点就是方程 ax² + bx + c = 0 的实数根。与 y 轴的交点为 c。

The vertex (turning point) has x-coordinate x = -b/2a. You can find the y-coordinate by substituting this x into the equation.

顶点（转折点）的 x 坐标为 x = -b/2a。将该 x 值代入方程即可求出 y 坐标。

Sketching a parabola requires three key features: the roots, the y-intercept, and the vertex. The axis of symmetry is the vertical line through the vertex.

绘制抛物线草图需要三个关键特征：根、y 轴截距和顶点。对称轴是过顶点的竖直线。

7. Roots and Turning Points | 根与顶点

The turning point of a quadratic function is also called the vertex. For y = a(x – h)² + k, the vertex is (h, k). This form is called vertex form.

二次函数的转折点也称为顶点。对于 y = a(x – h)² + k，顶点为 (h, k)。这种形式称为顶点式。

Completing the square converts standard form to vertex form. For example, y = x² + 4x + 5 can be written as y = (x + 2)² + 1, so the vertex is (-2, 1).

配方法将标准形式转换为顶点式。例如，y = x² + 4x + 5 可以写成 y = (x + 2)² + 1，因此顶点为 (-2, 1)。

The axis of symmetry is x = h. The maximum or minimum value of y is k.

对称轴为 x = h。y 的最大值或最小值为 k。

If the graph has two real roots, the x-coordinate of the vertex is exactly halfway between the two roots. This is a useful shortcut.

如果抛物线有两个实数根，那么顶点的 x 坐标恰好位于两根的正中间。这是一个有用的捷径。

8. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the following relationships hold:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，有以下关系：

α + β = -b/a, αβ = c/a

These formulas let you find the sum and product of the roots without solving the equation. They come from expanding a(x – α)(x – β).

这些公式使你不解方程就能求出根的和与积。它们来自于展开 a(x – α)(x – β)。

For example, for 2x² – 6x + 3 = 0, the sum of the roots is -(-6)/2 = 3, and the product is 3/2.

例如，对于 2x² – 6x + 3 = 0，根的和为 -(-6)/2 = 3，积为 3/2。

These relationships are also useful for constructing a quadratic with given roots. If the roots are p and q, then the equation is x² – (p + q)x + pq = 0.

这些关系也可用于构造具有给定根的二次方程。如果根为 p 和 q，则方程为 x² – (p + q)x + pq = 0。

9. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities can be solved by first finding the roots, then testing intervals on a number line or using the graph of the parabola.

解二次不等式时，先求出根，然后在数轴上测试区间，或借助抛物线图像判断。

For example, solve x² – x – 6 > 0. Factorise: (x – 3)(x + 2) > 0. The roots are -2 and 3.

例如，解 x² – x – 6 > 0。因式分解： (x – 3)(x + 2) > 0。根为 -2 和 3。

Since the parabola opens upwards, the expression is positive outside the interval between the roots. Therefore x < -2 or x > 3.

由于抛物线开口向上，表达式在两根之间的区间外为正。因此 x < -2 或 x > 3。

If the inequality were ≤ 0, the solution would be -2 ≤ x ≤ 3. Always consider whether the roots are included.

如果不等式是 ≤ 0，则解为 -2 ≤ x ≤ 3。务必考虑根是否包含在内。

10. Applications and Problem Solving | 应用与解题

Quadratic equations model many real situations, such as projectile motion, area problems, and profit maximisation.

二次方程可以模拟许多实际情况，例如抛体运动、面积问题和利润最大化。

Example: A rectangle has length 3 cm more than its width. Its area is 70 cm². Find the width.

例：一个长方形的长比宽多 3 cm，面积为 70 cm²。求宽。

Let width = w, length = w + 3. Then w(w + 3) = 70, so w² + 3w – 70 = 0.

设宽为 w，长为 w + 3。则 w(w + 3) = 70，即 w² + 3w – 70 = 0。

w² + 3w – 70 = 0 → (w + 10)(w – 7) = 0

Since w is positive, w = 7. Thus the width is 7 cm and the length is 10 cm.

由于 w 为正数，w = 7。因此宽为 7 cm，长为 10 cm。

When solving word problems, define the variable clearly, translate the conditions into an equation, solve it, and finally check whether your answers make sense in the real context.

解应用题时，要明确定义变量，将条件转化为方程，求解，最后检验答案在实际情境中是否合理。

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

One common mistake is forgetting to set the equation to zero before factorising or using the quadratic formula. Always rearrange ax² + bx + c = 0 first.

一个常见错误是忘记在因式分解或使用二次公式前将方程化为等于零的形式。务必先将方程整理为 ax² + bx + c = 0。

Another mistake is incorrectly simplifying the square root. For example, √12 must be written as 2√3, not 2√6. Practice simplifying surds.

另一个错误是化简根式出错。例如，√12 必须写成 2√3，而不是 2√6。多练习化简根式。

In exam questions, always show your working clearly. The method marks are often more valuable than the final answer.

在考试中，一定要清晰展示解题过程。方法分往往比最终答案更宝贵。

Remember to check the discriminant before deciding whether roots are real. Use the calculator only for arithmetic, not as a substitute for understanding.

在判断根是否为实数之前，记得先计算判别式。计算器只用于算术运算，不能替代对概念的理解。

12. Practice Questions | 练习题

To solidify your understanding, try these exam-style questions:

为巩固理解，请尝试以下考试风格的题目：

Solve x² – 7x + 12 = 0 by factorising.
解方程 x² – 7x + 12 = 0（用因式分解法）。
Solve 2x² + 3x – 5 = 0 using the quadratic formula.
用二次公式解 2x² + 3x – 5 = 0。
Find the vertex of y = x² – 6x + 11 by completing the square.
通过配方法求 y = x² – 6x + 11 的顶点。
Determine the nature of roots of 3x² – 2x + 5 = 0.
判断 3x² – 2x + 5 = 0 的根的性质。
Solve the inequality x² – 4x – 5 ≤ 0.
解不等式 x² – 4x – 5 ≤ 0。

Answers: 1) x = 3 or 4. 2) x = 1 or -2.5. 3) Vertex (3, 2). 4) No real roots. 5) -1 ≤ x ≤ 5.

答案：1) x = 3 或 4。2) x = 1 或 -2.5。3) 顶点 (3, 2)。4) 无实数根。5) -1 ≤ x ≤ 5。

Practice regularly, review your mistakes, and you will master quadratic equations in no time.

定期练习，反思错误，你很快就能掌握二次方程。

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Mastering Quadratic Equations | 二次方程完全指南