Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

Quadratic equations are a central topic in IGCSE Mathematics. Learning to solve them efficiently and understand their graphs is essential for higher-level algebra and many real-world applications.

二次方程是 IGCSE 数学的核心内容。掌握其求解技巧并理解对应图像，对后续代数学习及众多实际应用都至关重要。

1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。

Here, x represents an unknown variable. The highest power of x is 2, which is why the equation is called “quadratic” (from the Latin quadratus, meaning square).

这里 x 表示未知数，其最高次数为 2，因此被称为“二次”（quadratic，源自拉丁语 quadratus，意为“平方”）。

ax² is the quadratic term, with coefficient a.
ax² 是二次项，系数为 a。
bx is the linear term, with coefficient b.
bx 是一次项，系数为 b。
c is the constant term.
c 是常数项。

ax² + bx + c = 0, a ≠ 0

2. Solving by Factorisation | 因式分解法

If the quadratic expression can be factorised into two linear factors, we can set each factor equal to zero and solve.

如果二次式可以分解为两个一次因式的乘积，则可令每个因式等于零，从而求出解。

For example, solve x² – 5x + 6 = 0. The factors are (x – 2)(x – 3) = 0, so x = 2 or x = 3.

例如，解 x² – 5x + 6 = 0，因式分解为 (x – 2)(x – 3) = 0，所以 x = 2 或 x = 3。

Write the equation in the form ax² + bx + c = 0.
将方程写成 ax² + bx + c = 0 的形式。
Factorise the quadratic expression.
对二次式进行因式分解。
Set each factor equal to zero and solve the resulting linear equations.
令每个因式等于零，并解相应的一次方程。

This method works only when the expression can be factorised over integers or simple rationals.

此方法仅在二次式可以被分解为整数或简单有理数因子时才适用。

3. Solving by Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form (x + p)² + q, which can then be solved by taking square roots.

配方法将 ax² + bx + c 改写成 (x + p)² + q 的形式，然后通过开平方求解。

For example, solve x² + 6x + 4 = 0. Half of 6 is 3, so:

例如，解 x² + 6x + 4 = 0，6 的一半是 3，于是：

(x + 3)² – 9 + 4 = 0 → (x + 3)² = 5

Then x + 3 = ±√5, so x = -3 ± √5.

于是 x + 3 = ±√5，所以 x = -3 ± √5。

For a general quadratic ax² + bx + c = 0, divide by a first if a ≠ 1, then complete the square.

对于一般二次方程 ax² + bx + c = 0，若 a ≠ 1，则先除以 a，再进行配方。

Make sure the coefficient of x² is 1 (divide if necessary).
确保 x² 的系数为 1（必要时先除以该系数）。
Take half the coefficient of x, square it, then add and subtract this value.
取 x 系数的一半，平方后再加、减该值。
Rearrange into (x + p)² = k and solve for x.
整理成 (x + p)² = k 的形式，再解出 x。

4. The Quadratic Formula | 二次公式

The quadratic formula is a universal method that solves any quadratic equation, whether or not it has simple factors.

二次公式是通用的求解方法，适用于任何二次方程，无论其是否具有简单因子。

x = (-b ± √(b² – 4ac)) / 2a

For example, solve 2x² – 3x – 1 = 0. Here a = 2, b = -3, c = -1.

例如，解 2x² – 3x – 1 = 0，这里 a = 2，b = -3，c = -1。

x = (3 ± √((-3)² – 4·2·(-1))) / (2·2) = (3 ± √17) / 4

Always use this formula when factorisation is difficult or impossible. Ensure the equation is in the standard form before substituting.

当因式分解困难或无法分解时，始终使用该公式。代入前务必确保方程为标准形式。

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

The expression b² – 4ac is called the discriminant. It determines the number and type of roots.

表达式 b² – 4ac 称为判别式，它决定了根的个数与类型。

Discriminant 判别式	Nature of roots 根的形态
b² – 4ac > 0	Two distinct real roots 两个不同实数根
b² – 4ac = 0	One repeated real root 一个重根（两个相等实数根）
b² – 4ac < 0	No real roots 没有实数根

If the discriminant is a perfect square and a, b, c are rational, the equation factorises over rationals.

若判别式是完全平方数且 a、b、c 为有理数，则方程可以在有理数范围内因式分解。

6. The Graph of a Quadratic Function | 二次函数的图像

The graph of y = ax² + bx + c is a parabola. Its shape depends on the sign of a.

函数 y = ax² + bx + c 的图像是抛物线，其形状取决于 a 的正负。

If a > 0, the parabola opens upwards (U-shaped), and the vertex is a minimum point.
若 a > 0，抛物线开口向上（U 形），顶点为最小值点。
If a < 0, the parabola opens downwards (n-shaped), and the vertex is a maximum point.
若 a < 0，抛物线开口向下（∩ 形），顶点为最大值点。

The x-coordinate of the vertex is x = -b / 2a. The y-coordinate is found by substituting this x-value into the equation.

顶点的横坐标为 x = -b / 2a，纵坐标通过将该 x 值代入方程求得。

7. Roots, x-intercepts and the Axis of Symmetry | 根、x 轴交点与对称轴

The solutions of ax² + bx + c = 0 are the x-intercepts of the graph y = ax² + bx + c. They are also called the zeros or roots.

ax² + bx + c = 0 的解就是图像 y = ax² + bx + c 与 x 轴的交点，也称为零点或根。

The axis of symmetry of the parabola is the vertical line x = -b / 2a. It passes through the vertex and lies halfway between the two roots.

抛物线的对称轴是垂直线 x = -b / 2a，它经过顶点，并位于两个根的正中间。

If the equation has two roots r₁ and r₂, then:

若方程有两个根 r₁ 和 r₂，则：

x = -b / 2a = (r₁ + r₂) / 2

This relationship is useful for quickly finding the vertex or checking the accuracy of factorisation.

这一关系可快速求顶点坐标或检验因式分解是否正确。

8. Solving Quadratic Inequalities | 二次不等式的求解

Quadratic inequalities such as x² – x – 6 < 0 can be solved by sketching the parabola and determining where the graph is below or above the x-axis.

二次不等式（如 x² – x – 6 < 0）可通过画抛物线草图，确定图像在 x 轴下方或上方的区间来求解。

First, solve the corresponding equation x² – x – 6 = 0 to get critical points x = -2 and x = 3.

首先解对应方程 x² – x – 6 = 0，得到临界点 x = -2 和 x = 3。

Since the parabola opens upwards (a > 0), the graph is below the x-axis between the roots. Therefore x² – x – 6 < 0 means -2 < x < 3.

因为抛物线开口向上（a > 0），图像在两根之间位于 x 轴下方。所以 x² – x – 6 < 0 的解为 -2 < x < 3。

For > 0, the solution is x < r₁ or x > r₂ (outside the roots).
对于 > 0，解为 x < r₁ 或 x > r₂（两根之外）。
For < 0, the solution is r₁ < x < r₂ (between the roots).
对于 < 0，解为 r₁ < x < r₂（两根之间）。

9. Modelling Real-Life Situations | 实际情境建模

Quadratic equations often appear in physics, economics, and geometry problems. For example, the projectile height h after t seconds can be modelled by h = -4.9t² + vt + h₀.

二次方程常出现在物理、经济和几何问题中。例如，抛体在 t 秒后的高度 h 可建模为 h = -4.9t² + vt + h₀。

Example: A rectangular garden has an area of 60 m². Its length is 4 m longer than its width. Let the width be x m, then length is (x + 4) m, so:

例：一块矩形花园面积为 60 平方米，长比宽多 4 米。设宽为 x 米，则长为 (x + 4) 米，于是：

x(x + 4) = 60 → x² + 4x – 60 = 0

Solving by factorisation gives (x + 10)(x – 6) = 0, so x = 6 (since x = -10 is not a valid length).

因式分解得 (x + 10)(x – 6) = 0，故 x = 6（x = -10 不合题意，舍去）。

Always check that the answer makes sense in the context of the problem.

务必检查答案是否符合问题实际意义。

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

Students often lose marks on quadratic questions due to avoidable errors. Here are key tips.

学生在二次方程题目中常因可避免的错误丢分。以下是一些关键提示。

Always rearrange the equation to ax² + bx + c = 0 before factorising or using the formula.
在因式分解或套用公式前，务必先将方程整理为 ax² + bx + c = 0。
When using the quadratic formula, be careful with negative signs: write b = -3 as (-3)².
使用二次公式时，注意负号：如 b = -3 应写成 (-3)²。
If the coefficient of x² is negative, multiply the whole equation by -1 to make it positive.
如果 x² 系数为负，可将方程两边乘以 -1 使其变为正数。
When completing the square, remember to add (b/2)² and also subtract it to keep the equation balanced.
配方时，记得加上 (b/2)² 的同时也要减去它，以保持方程平衡。
For word problems, define variables clearly and discard any unrealistic negative solutions.
对于应用题，清晰定义变量，并舍去不切实际的负解。

Practising a variety of problems will build speed and confidence. Make sure you know when to use each method.

多做不同类型的题目能提升速度和信心，并确保你知道每种方法适用的时机。

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Solving Quadratic Equations | 二次方程求解