Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the core topics in IGCSE Mathematics. Understanding how to solve it — by factorising, using the quadratic formula, or completing the square — is essential for exams and for higher-level maths. This article covers all the key methods, with worked examples and common pitfalls to avoid.

二次方程是 IGCSE 数学的核心内容之一。掌握因式分解法、公式法和配方法等解法，对考试和后续数学学习都至关重要。本文将全面讲解关键解法、典型例题以及常见易错点。

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

A quadratic equation is a polynomial equation of degree 2. Its general form is: ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function y = ax² + bx + c is a parabola.

二次方程是次数为 2 的多项式方程，其一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。二次函数 y = ax² + bx + c 的图像是一条抛物线。

ax² + bx + c = 0

The value of a determines the direction of the parabola: if a > 0, it opens upward; if a < 0, it opens downward. The solutions of the equation are also called the roots, and they correspond to the x-intercepts of the parabola.

系数 a 决定抛物线的开口方向：若 a > 0，开口向上；若 a < 0，开口向下。方程的解也称为根，它们对应抛物线与 x 轴的交点。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic can be neatly factored. The idea is to rewrite the quadratic as a product of two linear factors, then set each factor equal to zero.

因式分解法通常是解决可分解二次方程的最快方法。思路是将二次式改写为两个一次因式的乘积，然后令每个因式等于零。

Write the equation in the form ax² + bx + c = 0.
Find two numbers whose sum is b and whose product is ac (when a ≠ 1, use grouping).
Rewrite the middle term and factor by grouping.
Set each bracket equal to 0 and solve for x.

将方程写成 ax² + bx + c = 0 的形式。

找到两个数，使它们的和为 b，积为 ac（当 a ≠ 1 时，可使用分组法）。

改写中间项并分组分解。

令每个括号等于 0，求出 x。

Example: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3

The root of the equation is obtained when either factor is zero. Always check the factors by expanding back to the original expression.

当任一因式为零时，即可得到方程的根。务必通过展开验证因式分解是否正确。

3. Solving by the Quadratic Formula | 公式法

When factorisation is difficult or impossible, use the quadratic formula. It works for any quadratic equation, including those with irrational or complex roots (at IGCSE level, roots are usually real).

当因式分解困难甚至无法分解时，可使用求根公式。它对任何二次方程都适用，包括根为无理数的情况（在 IGCSE 阶段，根通常为实数）。

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, identify a, b, and c from the equation ax² + bx + c = 0, then substitute them carefully. Be careful with negative signs.

使用公式时，从方程 ax² + bx + c = 0 中确定 a、b、c，然后代入。注意负号的处理。

Example: Solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2.

例如：解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。

x = (−3 ± √(9 − 4×2×(−2))) / (2×2) = (−3 ± √25) / 4 = (−3 ± 5) / 4

So x = 2/4 = 0.5 or x = −8/4 = −2. The quadratic formula is especially useful when the roots are not integers.

所以 x = 2/4 = 0.5 或 x = −8/4 = −2。当根不是整数时，公式法尤其有用。

4. Solving by Completing the Square | 配方法

Completing the square transforms a quadratic into the form (x + p)² = q. This method is also used to derive the quadratic formula and to find the vertex of a parabola.

配方法将二次方程转化为 (x + p)² = q 的形式。这种方法还可用于推导求根公式以及求抛物线的顶点。

Steps for solving x² + bx + c = 0:

解 x² + bx + c = 0 的步骤：

Move the constant term to the right side: x² + bx = −c.
Add (b/2)² to both sides.
Write the left side as a perfect square: (x + b/2)².
Take the square root of both sides and solve for x.

把常数项移到右边：x² + bx = −c。

两边加上 (b/2)²。

左边写成完全平方形式：(x + b/2)²。

两边开平方，解出 x。

Example: x² + 6x + 4 = 0 → (x + 3)² = 5 → x = −3 ± √5

If the coefficient of x² is not 1, first divide the whole equation by that coefficient.

如果 x² 的系数不是 1，先将整个方程除以该系数。

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

The expression b² − 4ac inside the quadratic formula is called the discriminant, often denoted by Δ (Delta). It tells us how many real roots the equation has.

求根公式中的 b² − 4ac 称为判别式，常用 Δ（德尔塔）表示。它告诉我们方程有多少个实数根。

If b² − 4ac > 0: two distinct real roots.
If b² − 4ac = 0: one repeated real root (a double root).
If b² − 4ac < 0: no real roots (two complex roots, not usually required at IGCSE).

若 b² − 4ac > 0：有两个不相等的实数根。

若 b² − 4ac = 0：有一个重根（二重根）。

若 b² − 4ac < 0：没有实数根（有两个复数根，通常不在 IGCSE 要求内）。

Δ = b² − 4ac

For example, for x² − 4x + 4 = 0, Δ = 16 − 16 = 0, so there is exactly one root: x = 2. Understanding the discriminant helps you decide which method to use.

例如，x² − 4x + 4 = 0 的判别式 Δ = 16 − 16 = 0，因此只有一个根：x = 2。理解判别式有助于选择合适的方法。

6. Word Problems Leading to Quadratics | 二次方程应用题

Many exam problems require you to form a quadratic equation from a real-world scenario, such as area, product of numbers, or projectile motion. Read the problem carefully and define the unknown variable.

许多考试题目需要从实际问题中建立二次方程，例如面积、数字乘积或抛体运动。仔细阅读题目，并定义未知数。

Example: The length of a rectangle is 3 cm more than its width, and its area is 70 cm². Find the width.

例：一个矩形的长比宽多 3 cm，面积是 70 cm²。求宽。

Let the width be x cm. Then length = x + 3 cm. The area is x(x + 3) = 70, so x² + 3x − 70 = 0. Factorise: (x + 10)(x − 7) = 0. Since x must be positive, x = 7.

设宽为 x cm，则长为 x + 3 cm。面积为 x(x + 3) = 70，即 x² + 3x − 70 = 0。因式分解得 (x + 10)(x − 7) = 0。因为 x 必须为正，所以 x = 7。

Always reject negative solutions if they do not make sense in the context.

如果负数解在情境中没有意义，要舍去。

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

The graph of y = ax² + bx + c is a parabola. The x-intercepts, if they exist, are the roots of the quadratic equation ax² + bx + c = 0. The y-intercept is the constant term c.

y = ax² + bx + c 的图像是抛物线。x 轴交点（若存在）就是二次方程 ax² + bx + c = 0 的根。y 轴截距为常数项 c。

The vertex (turning point) can be found by completing the square: y = a(x − h)² + k gives the vertex at (h, k). The axis of symmetry is the vertical line x = h.

通过配方可以找到顶点：y = a(x − h)² + k 的顶点为 (h, k)。对称轴是竖直线 x = h。

To sketch a quadratic graph, find the roots, the y-intercept, and the vertex. Also determine whether the parabola opens upward (a > 0) or downward (a < 0).

画二次函数草图时，需要找出根、y 轴截距和顶点。同时判断抛物线开口向上（a > 0）还是向下（a < 0）。

Example: y = x² − 4x + 3 = (x − 1)(x − 3) → x-intercepts at 1 and 3; vertex at (2, −1)

8. Special Cases and Rearranging | 特殊情况与变形

Sometimes the quadratic equation is not given in standard form. You must expand brackets, collect like terms, and set everything equal to zero before solving.

有时二次方程不是标准形式。你需要展开括号、合并同类项，并令方程等于零后再求解。

Example: Solve (x + 2)(x − 1) = 6. Expand first: x² + x − 2 = 6, so x² + x − 8 = 0. This does not factor simply; use the quadratic formula.

例：解 (x + 2)(x − 1) = 6。先展开：x² + x − 2 = 6，即 x² + x − 8 = 0。此式不易因式分解，使用公式法。

Also, be alert for equations with double roots, such as (x − 3)² = 0, which has only one root x = 3.

同时注意重根方程，如 (x − 3)² = 0，它只有一个根 x = 3。

9. Common Mistakes | 常见错误

Here are typical errors students make in exams, and how to avoid them.

以下是学生在考试中常犯的错误以及如何避免。

Forgetting to set the equation to zero before factorising.
Misidentifying a, b, c when a is negative.
Dropping the ± sign when taking square roots.
Making sign errors when substituting into the quadratic formula.
Not rejecting invalid solutions in word problems.

忘记在因式分解前将方程化为零。

当 a 为负数时，错误识别 a、b、c。

开平方时漏掉 ± 号。

代入求根公式时出现符号错误。

在应用题中没有舍去不合理的解。

Always double-check your answer by substituting it back into the original equation.

10. Practice Questions | 练习题

Try these questions to test your understanding.

尝试以下题目来检测你的理解。

Solve x² − 9 = 0.
Solve x² + 8x + 12 = 0 by factorisation.
Solve 2x² − 5x + 1 = 0 using the quadratic formula. Give answers to 2 decimal places.
Solve x² + 4x − 7 = 0 by completing the square.
The sum of two positive integers is 10, and the sum of their squares is 58. Find the integers.

解 x² − 9 = 0。

用因式分解法解 x² + 8x + 12 = 0。

用求根公式解 2x² − 5x + 1 = 0，答案保留两位小数。

用配方法解 x² + 4x − 7 = 0。

两个正整数的和为 10，它们的平方和为 58。求这两个整数。

Answers: (1) x = 3 or x = −3; (2) x = −2 or x = −6; (3) x ≈ 2.28 or x ≈ 0.22; (4) x = −2 ± √11; (5) The integers are 7 and 3.

答案：(1) x = 3 或 x = −3；(2) x = −2 或 x = −6；(3) x ≈ 2.28 或 x ≈ 0.22；(4) x = −2 ± √11；(5) 这两个整数是 7 和 3。

11. Summary | 总结

You now have three main methods for solving quadratic equations: factorisation (fastest when simple), the quadratic formula (works for everything), and completing the square (useful for finding the vertex). The discriminant helps you predict the number of roots. Always set the equation to zero first, and check your answers.

你现在掌握了三种解二次方程的主要方法：因式分解法（简单时最快）、公式法（适用于所有情况）和配方法（有助于求顶点）。判别式帮助预测根的个数。记得先将方程化为零，并检查答案。

Practice these methods with past paper questions, and you will be well prepared for the IGCSE exam.

通过历届试卷练习这些方法，你将为 IGCSE 考试做好充分准备。

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Solving Quadratic Equations | 解二次方程