📚 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。
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
e 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。
, 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
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
y 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。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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