Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations form one of the most important topics in IGCSE Mathematics. They appear throughout the syllabus, from algebra to graphs to problem-solving, and a strong command of quadratics is essential for exam success. In this article, we will explore every key method for solving quadratic equations, examine the discriminant, sketch quadratic graphs, and identify common pitfalls — all with clear worked examples.

二次方程是 IGCSE 数学中最重要的主题之一。它贯穿整个考纲，从代数到图像再到应用题，扎实掌握二次方程是考试成功的关键。本文将系统讲解解二次方程的每种核心方法，分析判别式，绘制二次函数图像，并指出常见易错点——均配有清晰的例题演示。

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

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Its general form is written as:

二次方程是次数为 2 的多项式方程，即变量的最高次数是 2。其一般形式为：

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are constants, and x is the unknown variable. The condition a ≠ 0 is essential — if a = 0, the equation becomes linear, not quadratic. The values of x that satisfy the equation are called the roots or solutions of the equation.

其中 a、b、c 是常数，x 是未知数。条件 a ≠ 0 至关重要——若 a = 0，方程就变成线性方程而非二次方程。满足方程的 x 值称为方程的根或解。

A quadratic equation can have at most two real roots. These roots can be distinct, equal, or even non-real (if the discriminant is negative, as we will see later). Graphically, the roots represent the x-intercepts of the parabola y = ax² + bx + c.

二次方程至多有两个实数根。这些根可以互不相同、相等，甚至不是实数（当判别式为负时，稍后讨论）。在图像上，根代表抛物线 y = ax² + bx + c 与 x 轴的交点。

2. Solving by Factorisation | 因式分解法

Factorisation is usually the fastest method when the quadratic has simple integer coefficients. The idea is to express the quadratic expression as a product of two linear factors, then use the zero product property: if pq = 0, then p = 0 or q = 0.

因式分解法通常是最快的方法，适用于系数为简单整数的二次方程。思路是将二次表达式写成两个一次因式的乘积，然后利用零积性质：若 pq = 0，则 p = 0 或 q = 0。

Example 1: Solve x² – 5x + 6 = 0.

例 1：解方程 x² – 5x + 6 = 0。

We look for two numbers whose product is 6 and whose sum is -5. The numbers are -2 and -3. Therefore:

我们要找到两个数，其乘积为 6，和为 -5。这两个数是 -2 和 -3。因此：

(x – 2)(x – 3) = 0

So x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

所以 x – 2 = 0 或 x – 3 = 0，解得 x = 2 或 x = 3。

When the coefficient of x² is not 1, we can use the method of grouping. For example, solving 2x² + 7x + 3 = 0: multiply a and c to get 6, then find two numbers that multiply to 6 and add to 7 — these are 6 and 1. Rewrite the middle term: 2x² + 6x + x + 3 = 0, then factor by grouping:

当 x² 的系数不是 1 时，可以使用分组分解法。例如，解 2x² + 7x + 3 = 0：将 a 和 c 相乘得 6，然后找到两个数相乘为 6、相加为 7——即 6 和 1。重写中间项：2x² + 6x + x + 3 = 0，再分组分解：

2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0

Hence x = -½ or x = -3.

因此 x = -½ 或 x = -3。

Always check whether the equation is in standard form before factorising. If it is not, rearrange all terms to one side first.

在因式分解前务必确认方程是否已化为标准形式。如果不是，先将所有项移到等号一侧。

3. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that transforms a quadratic expression into the form a(x + p)² + q. This method is especially useful when factorisation is not straightforward, and it also helps in deriving the quadratic formula and solving equations.

配方法是一种强有力的技巧，它把二次表达式转化为 a(x + p)² + q 的形式。当因式分解不易直接进行时，此方法尤为实用，同时也是推导二次公式和求解方程的基础。

Example 2: Solve x² + 6x + 2 = 0 by completing the square.

例 2：用配方法解 x² + 6x + 2 = 0。

Step 1: Take the coefficient of x, which is 6, halve it to get 3, and square it to get 9. Add and subtract 9:

步骤 1：取 x 的系数 6，一半为 3，平方得 9。加 9 再减 9：

x² + 6x + 9 – 9 + 2 = 0

Step 2: Rewrite as (x + 3)² – 7 = 0.

步骤 2：改写为 (x + 3)² – 7 = 0。

(x + 3)² = 7

Step 3: Take the square root of both sides:

步骤 3：两边开平方：

x + 3 = ±√7

Therefore x = -3 + √7 or x = -3 – √7. In decimal form, x ≈ 0.646 or x ≈ -6.646.

因此 x = -3 + √7 或 x = -3 – √7。用小数表示，x ≈ 0.646 或 x ≈ -6.646。

If the coefficient of x² is not 1, first factor it out before completing the square. For instance, 2x² – 8x + 5 = 0 can be written as 2(x² – 4x) + 5 = 0, then complete the square inside the brackets.

若 x² 的系数不是 1，先将其提出来再配方。例如 2x² – 8x + 5 = 0 可写成 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 universal method to solve any quadratic equation. For the equation ax² + bx + c = 0, the roots are given by:

二次公式是解任意二次方程的通用方法。对于方程 ax² + bx + c = 0，其根为：

x = (-b ± √(b² – 4ac)) / 2a

This formula is derived by completing the square on the general quadratic. It works for all quadratics, including those with irrational or complex roots.

该公式通过对一般二次方程配方推导得出。它对所有二次方程都适用，包括有無理数根或复数根的情况。

Example 3: Solve 3x² – 5x – 2 = 0 using the quadratic formula.

例 3：用二次公式解 3x² – 5x – 2 = 0。

Here a = 3, b = -5, c = -2. Substitute into the formula:

这里 a = 3，b = -5，c = -2。代入公式：

x = (5 ± √(25 – 4 × 3 × (-2))) / (2 × 3) = (5 ± √49) / 6 = (5 ± 7) / 6

So x = (5 + 7)/6 = 2, or x = (5 – 7)/6 = -1/3.

所以 x = (5 + 7)/6 = 2，或 x = (5 – 7)/6 = -1/3。

Always substitute carefully and remember that -b means the negative of b. A common error is forgetting the negative sign when b is already negative.

代入时要格外小心，注意 -b 表示 b 的相反数。一个常见错误是在 b 本身为负时忘记符号。

5. The Discriminant | 判别式

The expression b² – 4ac inside the quadratic formula is called the discriminant, often denoted by Δ. It determines the nature of the roots without actually solving the equation.

二次公式中根号下的表达式 b² – 4ac 称为判别式，通常用 Δ 表示。它无需解方程即可判断根的性质。

Value of Δ	Nature of Roots	Graphical Interpretation
Δ > 0	Two distinct real roots	Parabola crosses the x-axis at two points
Δ = 0	Two equal real roots (one repeated root)	Parabola touches the x-axis at one point
Δ < 0	No real roots (complex roots)	Parabola does not intersect the x-axis

Example 4: Determine the nature of the roots of x² + 4x + 5 = 0.

例 4：判断方程 x² + 4x + 5 = 0 的根的性质。

Here Δ = 4² – 4 × 1 × 5 = 16 – 20 = -4, which is negative. Therefore the equation has no real roots.

这里 Δ = 4² – 4 × 1 × 5 = 16 – 20 = -4，为负数。因此该方程没有实数根。

The discriminant also helps in problems involving conditions on parameters. For example, if the equation kx² + 6x + 3 = 0 has two equal roots, then Δ = 0, giving 36 – 12k = 0, so k = 3.

判别式还用于含参数的题目。例如，若方程 kx² + 6x + 3 = 0 有两相等实根，则 Δ = 0，即 36 – 12k = 0，解得 k = 3。

6. Sketching Quadratic Graphs | 绘制二次函数图像

The graph of a quadratic function y = ax² + bx + c is a parabola. The sign of a determines the direction of the curve: if a > 0, the parabola opens upward; if a < 0, it opens downward.

二次函数 y = ax² + bx + c 的图像是抛物线。a 的符号决定开口方向：a > 0 时开口向上；a < 0 时开口向下。

To sketch a quadratic graph accurately, identify the following key features:

要准确绘制二次函数图像，需要确定以下几个关键特征：

The y-intercept: the value of c, which is the point where the graph crosses the y-axis.
The x-intercepts: the roots of ax² + bx + c = 0, if they exist.
The vertex (turning point): its x-coordinate is -b/2a, and the y-coordinate is found by substituting this value into the equation.
The axis of symmetry: the vertical line x = -b/2a passing through the vertex.

y 轴截距：即 c 的值，是图像与 y 轴的交点。
x 轴截距：方程 ax² + bx + c = 0 的根（若存在）。
顶点（转折点）：其 x 坐标为 -b/2a，y 坐标通过将该值代入方程求得。
对称轴：通过顶点的竖直直线 x = -b/2a。

Example 5: Sketch y = x² – 2x – 3.

例 5：画出 y = x² – 2x – 3 的图像。

The y-intercept is -3. Factorising x² – 2x – 3 = (x – 3)(x + 1) gives roots x = 3 and x = -1. The x-coordinate of the vertex is -(-2)/(2 × 1) = 1, and y(1) = 1 – 2 – 3 = -4. The vertex is at (1, -4).

y 轴截距为 -3。因式分解 x² – 2x – 3 = (x – 3)(x + 1) 得到根 x = 3 和 x = -1。顶点的 x 坐标为 -(-2)/(2 × 1) = 1，y(1) = 1 – 2 – 3 = -4。顶点为 (1, -4)。

Since a = 1 > 0, the parabola opens upward. With the vertex, intercepts and symmetry, we can sketch the graph accurately.

由于 a = 1 > 0，抛物线开口向上。结合顶点、截距和对称性，即可准确画出图像。

7. 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：

Sum of roots: α + β = -b/a

Product of roots: α × β = c/a

These formulas allow us to form a quadratic equation from its roots without solving any equation. For instance, if the roots are 4 and -2, then the sum is 2 and the product is -8, so the equation is x² – 2x – 8 = 0.

这些公式让我们无需解方程即可由根构造二次方程。例如，若根为 4 和 -2，则和为 2，积为 -8，所以方程为 x² – 2x – 8 = 0。

This is also useful when checking solutions. After solving a quadratic equation, verify that the sum and product of your answers match -b/a and c/a respectively.

这在检验答案时同样有用。解完二次方程后，验证两根之和与两根之积是否分别等于 -b/a 和 c/a。

8. Applications of Quadratic Equations | 二次方程的应用

Quadratic equations arise naturally in many real-life contexts, including area problems, projectile motion, economic profit models, and geometry.

二次方程在现实生活中广泛出现，包括面积问题、抛体运动、经济利润模型和几何问题等。

Example 6: A rectangular garden has a length that is 3 metres longer than its width. If the area is 40 m², find the dimensions of the garden.

例 6：一块矩形花园的长比宽长 3 米。若面积为 40 平方米，求花园的尺寸。

Let the width be x metres. Then the length is x + 3 metres. The area is x(x + 3) = 40. Expanding gives x² + 3x – 40 = 0. Factorising: (x + 8)(x – 5) = 0, so x = 5 (rejecting x = -8 as a negative width is not meaningful). The garden is 5 m by 8 m.

设宽为 x 米，则长为 x + 3 米。面积为 x(x + 3) = 40。展开得 x² + 3x – 40 = 0。因式分解：(x + 8)(x – 5) = 0，所以 x = 5（舍去 x = -8，因为宽度不能为负）。花园的尺寸为 5 米 × 8 米。

In physics, the height of an object thrown upward can be modelled by h = -5t² + vt + h₀, where t is time, v is initial velocity, and h₀ is the initial height. Setting h = 0 allows us to find when the object lands.

在物理中，物体上抛的高度可以用 h = -5t² + vt + h₀ 建模，其中 t 是时间，v 是初速度，h₀ 是初始高度。令 h = 0 即可求物体落地的时间。

9. Common Mistakes and How to Avoid Them | 常见错误与应对策略

Students often lose marks on quadratic equations due to avoidable errors. Below are the most frequent pitfalls and their solutions.

学生在二次方程题目中常因可避免的错误而失分。以下是最常见的陷阱及其对策。

Common Mistake	Correct Approach
Dividing both sides by x, losing the root x = 0	Factorise instead: x(x + a) = 0, giving x = 0 as a valid root
Forgetting to rearrange to ax² + bx + c = 0 before factorising	Always bring all terms to one side first and simplify
Misapplying the quadratic formula when a is negative	Multiply the whole equation by -1 first, or substitute signs carefully
Omitting the ± symbol when taking square roots	Remember that √n only gives the positive root; always write ±√n
Using the quadratic formula instead of completing the square when instructed	Read the question carefully and follow the required method

Another common issue is sign errors when expanding brackets. For example, (x – 5)² is x² – 10x + 25, not x² – 25. Always expand carefully using the standard identity (a ± b)² = a² ± 2ab + b².

另一个常见问题是展开括号时的符号错误。例如 (x – 5)² 等于 x² – 10x + 25，而不是 x² – 25。务必使用标准恒等式 (a ± b)² = a² ± 2ab + b² 仔细展开。

10. Practice Questions | 练习题

Mastery comes with practice. Try the following questions on your own before checking the answers.

熟练掌握来自练习。先独立尝试以下题目，再对照答案自查。

Question 1: Solve x² – 7x + 12 = 0 by factorisation.

练习 1：用因式分解法解 x² – 7x + 12 = 0。

Question 2: Solve x² + 4x – 9 = 0 by completing the square, giving your answers correct to 2 decimal places.

练习 2：用配方法解 x² + 4x – 9 = 0，答案精确到小数点后两位。

Question 3: Use the quadratic formula to solve 2x² – 3x – 1 = 0.

练习 3：用二次公式解 2x² – 3x – 1 = 0。

Question 4: Find the value of k such that kx² + 4x + 2 = 0 has one repeated root.

练习 4：求 k 的值，使得方程 kx² + 4x + 2 = 0 有一个重根。

Question 5: A right-angled triangle has an area of 30 cm². If its height exceeds its base by 7 cm, find the base length.

练习 5：一个直角三角形的面积为 30 平方厘米。若其高比底边长 7 厘米，求底边的长度。

Answers: (1) x = 3 or x = 4; (2) x ≈ 1.61 or x ≈ -5.61; (3) x ≈ 1.78 or x ≈ -0.28; (4) k = 2; (5) base = 5 cm.

答案：(1) x = 3 或 x = 4；(2) x ≈ 1.61 或 x ≈ -5.61；(3) x ≈ 1.78 或 x ≈ -0.28；(4) k = 2；(5) 底边 = 5 厘米。

11. Quick Revision Checklist | 快速复习清单

Use this checklist to confirm that you have mastered each aspect of quadratic equations before entering the exam hall.

进入考场前，请用这份清单确认自己已完全掌握二次方程的各个方面。

I can identify a quadratic equation in standard form ax² + bx + c = 0.
I can solve by factorisation, including cases where a ≠ 1.
I can solve by completing the square and express answers in surd form.
I can substitute into the quadratic formula accurately.
I can use the discriminant Δ = b² – 4ac to determine the nature of roots.
I can sketch a parabola using intercepts, vertex and axis of symmetry.
I can apply the sum (α + β = -b/a) and product (αβ = c/a) of roots.
I can translate word problems into quadratic equations and interpret the solutions.

我能识别标准形式 ax² + bx + c = 0 的二次方程。
我能用因式分解法求解，包括 a ≠ 1 的情形。
我能用配方法求解，并能以根式形式表达答案。
我能准确代入二次公式计算。
我能用判别式 Δ = b² – 4ac 判断根的性质。
我能利用截距、顶点和对称轴绘制抛物线。
我能运用根的和（α + β = -b/a）与积（αβ = c/a）公式。
我能将文字应用题转化为二次方程并解释所得解。

If you can tick every box, you are well prepared for the quadratics questions in your IGCSE examination. Remember that consistent practice is the key to retaining these skills.

如果你能勾选以上所有项目，说明你已为 IGCSE 考试中的二次方程题目做好了充分准备。记住，持续练习是掌握这些技能的关键。

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Solving Quadratic Equations | 解二次方程