Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations form one of the most important topics in IGCSE Mathematics. They appear in algebra, geometry, graphs, and real-world problem solving. This revision guide covers every key skill you need, from factorisation to the quadratic formula, with clear explanations in English and Chinese.

二次方程是 IGCSE 数学中最核心的主题之一。它贯穿代数、几何、图像以及实际应用题。本复习指南涵盖所有关键技能，从因式分解到求根公式，并提供中英双语清晰讲解。

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

A quadratic equation is an equation where the highest power of the variable is 2. The standard form is written as ax² + bx + c = 0, where a, b and c are real numbers, and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

二次方程是指变量的最高次数为 2 的方程。其标准形式写作 ax² + bx + c = 0，其中 a、b、c 为实数，且 a ≠ 0。若 a = 0，方程就变成线性方程，而不是二次方程。

The coefficient a is the quadratic coefficient.
The coefficient b is the linear coefficient.
The constant term is c.

a 是二次项系数。
b 是一次项系数。
c 是常数项。

ax² + bx + c = 0 (a ≠ 0)

For example, 2x² – 5x + 3 = 0 is a quadratic equation with a = 2, b = –5 and c = 3. The equation x² = 9 is also quadratic, because it can be rewritten as x² + 0x – 9 = 0.

例如，2x² – 5x + 3 = 0 是一个二次方程，其中 a = 2，b = –5，c = 3。方程 x² = 9 也是二次方程，因为它可以改写为 x² + 0x – 9 = 0。

2. Standard Form and Key Vocabulary | 标准形式与关键术语

Before solving a quadratic equation, you must rearrange it so that one side equals zero. This is called the standard form. For example, x² = 4x – 3 must be rewritten as x² – 4x + 3 = 0.

在解二次方程之前，必须将方程整理为一边等于零的形式，即标准形式。例如，x² = 4x – 3 必须改写为 x² – 4x + 3 = 0。

A root (or solution) is a value of x that makes the equation true.
The solution set is the set of all roots.
An equation may have two distinct roots, one repeated root, or no real roots.

根（或解）是使方程成立的 x 的值。
解集是所有根的集合。
一个方程可能有两个不同实根、一个重根，或没有实根。

When you expand (x – 2)(x + 3), you get x² + x – 6. Therefore, the equation x² + x – 6 = 0 can be written as (x – 2)(x + 3) = 0. This is called the factored form.

展开 (x – 2)(x + 3) 后得到 x² + x – 6。因此，方程 x² + x – 6 = 0 可以写成 (x – 2)(x + 3) = 0。这称为因式分解形式。

Factored form: (x – p)(x – q) = 0

3. Solving by Factorisation | 因式分解法

Factorisation is often the quickest way to solve a quadratic equation when the coefficients are simple integers. If you can write the quadratic as a product of two linear factors, the roots can be found immediately.

当系数为简单整数时，因式分解通常是解二次方程最快的方法。如果你能把二次式写成两个一次因式的乘积，就可以立即求出根。

Solve x² – 5x + 6 = 0 by factorisation. We look for two numbers whose product is 6 and whose sum is –5. The numbers are –2 and –3.

用因式分解法解 x² – 5x + 6 = 0。我们要找两个数，其乘积为 6，和为 –5。这两个数是 –2 和 –3。

(x – 2)(x – 3) = 0

If the product of two factors is zero, then at least one factor must be zero. So x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

如果两个因式的乘积为零，则至少有一个因式为零。所以 x – 2 = 0 或 x – 3 = 0，因此 x = 2 或 x = 3。

First write the equation in standard form.
Find factor pairs of c that add to b.
Set each factor equal to zero and solve.

首先将方程写成标准形式。
找到常数 c 的因数对，使其和为 b。
令每个因式等于零并求解。

If the coefficient a is not 1, you may need to factor by grouping or use trial and improvement. For example, 2x² + 7x + 3 = (2x + 1)(x + 3).

如果 a 不为 1，则可能需要分组分解或试凑。例如，2x² + 7x + 3 = (2x + 1)(x + 3)。

4. Solving by Completing the Square | 配方法

Completing the square is a method that rewrites a quadratic expression in the form a(x + h)² + k. This is especially useful when factorisation is difficult or when you need to find the vertex of a parabola.

配方法将二次表达式改写为 a(x + h)² + k 的形式。当因式分解困难或需要求抛物线顶点时，这种方法尤其有用。

Start with x² + 6x + 5 = 0. Take half of 6, which is 3, and square it to get 9. Rewrite the first two terms as (x + 3)² – 9 + 5 = 0.

从 x² + 6x + 5 = 0 开始。取 6 的一半为 3，平方得到 9。将前两项改写为 (x + 3)² – 9 + 5 = 0。

x² + 6x + 5 = (x + 3)² – 4

Now solve (x + 3)² – 4 = 0. Add 4 to both sides: (x + 3)² = 4. Taking square roots gives x + 3 = ±2, so x = –1 or x = –5.

现在解 (x + 3)² – 4 = 0。两边加 4： (x + 3)² = 4。开平方得 x + 3 = ±2，所以 x = –1 或 x = –5。

Divide the equation by a if a ≠ 1.
Use (b/2)² to complete the square.
Always keep the equation balanced.

如果 a ≠ 1，先将方程除以 a。
用 (b/2)² 配平方。
始终保持方程平衡。

Completing the square also reveals the minimum or maximum value of the quadratic expression. For (x + 3)² – 4, the minimum value is –4 when x = –3.

配方法还能揭示二次表达式的最小值或最大值。对于 (x + 3)² – 4，当 x = –3 时，最小值为 –4。

5. Solving by the Quadratic Formula | 公式法

The quadratic formula is a universal method that works for any quadratic equation. You can use it when factorisation is impossible or too time-consuming.

求根公式是适用于任何二次方程的通用方法。当因式分解不可能或太费时时，可以使用它。

x = ( –b ± √(b² – 4ac) ) / (2a)

For the equation 2x² – 4x – 3 = 0, we have a = 2, b = –4, c = –3. Substitute these values into the formula.

对于方程 2x² – 4x – 3 = 0，有 a = 2，b = –4，c = –3。将这些值代入公式。

x = ( 4 ± √(16 + 24) ) / 4 = ( 4 ± √40 ) / 4

Simplify √40 as 2√10, so x = (4 ± 2√10) / 4 = 2 ± √10 / 2. These are two irrational roots.

将 √40 化简为 2√10，所以 x = (4 ± 2√10) / 4 = 2 ± √10 / 2。这是两个无理数根。

Identify a, b and c correctly.
Be careful with negative signs.
Simplify the square root if possible.

正确识别 a、b 和 c。
注意负号。
尽量化简根号。

The quadratic formula is also used to solve equations with irrational or complex roots, though complex roots are not required at IGCSE level.

求根公式也可用于解无理数根或复数根，但 IGCSE 阶段不要求复数根。

6. The Discriminant | 判别式

The discriminant is the expression b² – 4ac inside the square root of the quadratic formula. It tells us the number and type of roots without solving the equation completely.

判别式是求根公式中根号内的表达式 b² – 4ac。它能告诉我们根的个数和类型，而无需完全解方程。

Δ = b² – 4ac

If Δ > 0, there are two distinct real roots.
If Δ = 0, there is one repeated real root.
If Δ < 0, there are no real roots.

如果 Δ > 0，有两个不同的实根。
如果 Δ = 0，有一个重根（重复实根）。
如果 Δ < 0，没有实根。

For example, the equation x² – 4x + 4 = 0 has Δ = 16 – 16 = 0. Therefore, it has exactly one root, x = 2. The graph just touches the x-axis at one point.

例如，x² – 4x + 4 = 0 的 Δ = 16 – 16 = 0。因此它只有一个根 x = 2。图像与 x 轴仅相切于一点。

The discriminant is also useful when a problem states that a quadratic equation has equal roots. Then you can set b² – 4ac = 0 and solve for the unknown parameter.

当题目说明二次方程有等根时，判别式也很有用。此时可令 b² – 4ac = 0，并求解未知参数。

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

The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upward and has a minimum point. When a < 0, it opens downward and has a maximum point.

y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，有最低点；当 a < 0 时，开口向下，有最高点。

The y-intercept is the point (0, c).
The x-intercepts are the real roots of ax² + bx + c = 0.
The axis of symmetry is a vertical line passing through the vertex.

y 轴截距是点 (0, c)。
x 轴截距是 ax² + bx + c = 0 的实根。
对称轴是经过顶点的一条竖直线。

To sketch a quadratic graph, first find the roots, then the y-intercept, and then the vertex. For example, y = x² – 2x – 3 has roots x = 3 and x = –1, and y-intercept (0, –3). The axis of symmetry is x = 1, because it is halfway between –1 and 3.

要画二次函数草图，先求根，再求 y 截距，然后求顶点。例如，y = x² – 2x – 3 的根为 x = 3 和 x = –1，y 截距为 (0, –3)。对称轴是 x = 1，因为它是 –1 和 3 的中点。

Substitute x = 1 into y to find the vertex: y = 1 – 2 – 3 = –4. So the vertex is (1, –4). The parabola opens upward because a = 1 > 0.

将 x = 1 代入 y： y = 1 – 2 – 3 = –4。因此顶点为 (1, –4)。由于 a = 1 > 0，抛物线开口向上。

8. Turning Points and Symmetry | 顶点与对称轴

The turning point of a parabola is its highest or lowest point. For a quadratic written as y = a(x – h)² + k, the turning point is (h, k). The axis of symmetry is x = h.

抛物线的顶点是它的最高点或最低点。对于写成 y = a(x – h)² + k 的二次函数，顶点为 (h, k)，对称轴为 x = h。

Alternatively, for y = ax² + bx + c, the x-coordinate of the vertex is given by x = –b / (2a). Then substitute to find the y-coordinate.

另外，对于 y = ax² + bx + c，顶点的 x 坐标为 x = –b / (2a)，然后代入求 y 坐标。

xₖ = –b / (2a)

Using the previous example y = x² – 2x – 3, we have a = 1, b = –2, so x = 2 / 2 = 1. Then y = 1 – 2 – 3 = –4. The turning point is (1, –4).

用前面的例子 y = x² – 2x – 3，a = 1，b = –2，所以 x = 2 / 2 = 1。然后 y = 1 – 2 – 3 = –4。顶点为 (1, –4)。

If a > 0, the turning point is a minimum.
If a < 0, the turning point is a maximum.
The vertex always lies on the axis of symmetry.

如果 a > 0，顶点是最小值。
如果 a < 0，顶点是最大值。
顶点总是在对称轴上。

Completing the square is the easiest way to identify the turning point directly from the equation.

配方法是从方程中直接确定顶点的最简单方法。

9. Applications in Problem Solving | 应用题

Quadratic equations model many real-life situations, such as projectile motion, area problems, and profit maximisation. You often need to translate a word problem into a quadratic equation.

二次方程可以建模许多现实情境，如抛体运动、面积问题、利润最大化等。通常你需要将文字题转化为二次方程。

Example: The length of a rectangle is 3 cm longer than its width. The area is 40 cm². Find its dimensions.

例：一个矩形的长比宽长 3 cm，面积为 40 cm²。求它的尺寸。

Let the width be x cm. Then the length is x + 3 cm. The area is x(x + 3) = 40. Expanding gives x² + 3x – 40 = 0.

设宽为 x cm，则长为 x + 3 cm。面积 x(x + 3) = 40。展开得 x² + 3x – 40 = 0。

x² + 3x – 40 = 0

Factorise: (x + 8)(x – 5) = 0, so x = –8 or x = 5. Since width cannot be negative, x = 5. The length is 8 cm.

因式分解： (x + 8)(x – 5) = 0，所以 x = –8 或 x = 5。由于宽度不能为负，x = 5。长为 8 cm。

Define a variable for the unknown quantity.
Set up an equation from the given conditions.
Check whether any roots are impossible in context.

为未知量设变量。
根据条件建立方程。
检查是否有在当前情境下不合适的根。

Another common application is finding the maximum height of an object thrown upwards, where the height h(t) is given by a quadratic function of time t.

另一个常见应用是求抛向空中的物体的最大高度，其中高度 h(t) 是时间 t 的二次函数。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

Many students lose marks on quadratics because of small errors in rearrangement, signs, or arithmetic. Here is how to avoid them in your exam.

许多学生在二次方程上失分是因为整理、符号或算术方面的小错误。以下是在考试中避免这些错误的方法。

Always write the equation in the form ax² + bx + c = 0 before solving.
Do not cancel a factor of x if x = 0 is a possible solution.
When using the quadratic formula, check that a is the coefficient of x², not just the number before x² after division.
Simplify square roots fully, e.g. √12 = 2√3.

求解前总是将方程整理为 ax² + bx + c = 0 的形式。
如果 x = 0 可能是解，则不要随意约去 x。
使用求根公式时，确认 a 是 x² 的系数，而不是除法后的其他数。
完全化简根号，例如 √12 = 2√3。

Check answers by substituting back into the original equation.
For graphical questions, show the coordinates of the vertex and intercepts clearly.
If a quadratic has no real roots, write “no real roots” rather than leaving a blank.

通过代回原方程检查答案。
对于图像题，清晰标出顶点和截距的坐标。
如果二次方程没有实根，应写上 “无实根”，而不是留空。

Practise past paper questions every day. The more you practise, the faster you will recognise which method to use for each equation.

每天练习往年真题。你练习得越多，就能越快识别每个方程该用哪种方法。

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Mastering Quadratic Equations | 掌握二次方程