📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from basic factorisation to graph sketching and real-world modelling. A strong grasp of these techniques is essential for both Core and Extended papers, especially when questions combine algebra, graphs, and problem solving. This article explains the standard form, four main solution methods, the discriminant, graph features, and common exam applications. Work through each section carefully, then use the practice questions to test your understanding.
二次方程贯穿 IGCSE 数学课程的始终，从基础因式分解到图像绘制和实际问题建模。熟练掌握这些技巧对 Core 和 Extended 试卷都至关重要，尤其是当题目综合考查代数、图像和应用题时。本文将讲解二次方程的标准形式、四种主要解法、判别式、图像特征以及常见考试应用。请认真学习每一节，然后用练习题检验自己的理解。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation that can be written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The term ax² is the quadratic term, bx is the linear term, and c is the constant term. If a = 0, the equation becomes linear, so this condition is always required.
二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数，且 a ≠ 0。ax² 是二次项，bx 是一次项，c 是常数项。如果 a = 0，方程就变成一次方程，因此这个条件始终要满足。
For example, x² - 5x + 6 = 0, 2x² + 3x - 2 = 0 and -x² + 4 = 0 are all quadratic equations. The values of x that make the equation true are called the roots or solutions of the equation. A quadratic equation can have two real roots, one repeated real root, or no real roots.
例如，x² - 5x + 6 = 0、2x² + 3x - 2 = 0 和 -x² + 4 = 0 都是二次方程。使方程成立的 x 值称为方程的根或解。二次方程可以有两个实根、一个重根，或者没有实根。
2. Standard Form and Key Terms | 标准形式与关键术语
Before solving any quadratic equation, rewrite it in the standard form ax² + bx + c = 0. This means expanding brackets, collecting like terms, and making one side equal to zero. For example, x(x - 3) = 4 becomes x² - 3x - 4 = 0 after expansion and rearranging.
解任何二次方程之前，都要先把它整理成标准形式 ax² + bx + c = 0。这意味着要先展开括号、合并同类项，并让方程的一边等于零。例如，x(x - 3) = 4 经过展开和移项后变为 x² - 3x - 4 = 0。
Key terms include the leading coefficient a, the constant term c, and the roots or solutions. The degree of the equation is 2 because the highest power of x is 2. In IGCSE, coefficients are usually integers or simple fractions, but surds and decimals can appear in later questions.
关键术语包括首项系数 a、常数项 c 以及根或解。方程的次数是 2，因为 x 的最高次数是 2。在 IGCSE 中，系数通常是整数或简单分数，但在较难的题目中也可能出现根式和小数。
Always check that the equation is fully simplified before choosing a method. Some equations may look complicated but reduce quickly to x² + 5x - 14 = 0 or 3x² - 12 = 0.
在选择解法之前，一定要确认方程已经完全化简。有些方程看起来复杂，但很快就能化简为 x² + 5x - 14 = 0 或 3x² - 12 = 0。
3. Solving by Factorisation | 因式分解法
Factorisation is usually the fastest method when the quadratic expression has rational roots. For a monic quadratic x² + bx + c = 0, find two numbers whose sum is b and whose product is c. Then write x² + bx + c = (x + p)(x + q), where p + q = b and pq = c.
当二次式有有理根时，因式分解通常是最快的方法。对于首一二次式 x² + bx + c = 0，找出两个数，使它们的和等于 b，乘积等于 c。然后写成 x² + bx + c = (x + p)(x + q)，其中 p + q = b，pq = c。
For example, to solve x² - 7x + 12 = 0, find two numbers that add to -7 and multiply to 12. Those numbers are -3 and -4, so:
例如，解 x² - 7x + 12 = 0，要找出两个数，它们的和是 -7，乘积是 12。这两个数是 -3 和 -4，因此：
x² - 7x + 12 = (x - 3)(x - 4) = 0
Set each factor equal to zero: x - 3 = 0 gives x = 3, and x - 4 = 0 gives x = 4. The solutions are x = 3 and x = 4.
令每个因式为零：x - 3 = 0 得 x = 3，x - 4 = 0 得 x = 4。解为 x = 3 和 x = 4。
When a ≠ 1, use the ac method. For 2x² + 5x - 3 = 0, multiply a × c = 2 × (-3) = -6. Find two numbers that multiply to -6 and add to 5, namely 6 and -1. Split the middle term and factor in pairs:
当 a ≠ 1 时，使用 ac 法。对于 2x² + 5x - 3 = 0，计算 a × c = 2 × (-3) = -6。找出两个数，它们的乘积是 -6，和是 5，即 6 和 -1。拆分一次项，然后分组分解：
2x² + 6x - x - 3 = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3) = 0
The roots are x = 1/2 and x = -3. Always verify by substituting back into the original equation.
根为 x = 1/2 和 x = -3。一定要代回原方程进行验证。
4. Solving by Completing the Square | 配方法
Completing the square is a reliable method that works for any quadratic equation. It is especially useful when the equation does not factorise easily. The key identity is:
配方法是一种可靠的方法，适用于任何二次方程。当方程不容易因式分解时，它特别有用。关键恒等式为：
x² + bx = (x + b/2)² - (b/2)²
To solve x² + 6x + 2 = 0, first move the constant term to the right:
解 x² + 6x + 2 = 0 时，先把常数项移到右边：
x² + 6x = -2
Add (6/2)² = 9 to both sides:
两边同时加上 (6/2)² = 9：
x² + 6x + 9 = 7
(x + 3)² = 7
Take the square root of both sides, remembering both positive and negative roots:
两边开平方，同时记住正负两个根：
x + 3 = ±√7
x = -3 ± √7
The solutions are x = -3 + √7 and x = -3 - √7. Completing the square also gives the turning point of a quadratic graph, which is covered later.
解为 x = -3 + √7 和 x = -3 - √7。配方法还可以给出二次函数图像的顶点，后面会讲到。
5. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0 by direct substitution:
求根公式通过直接代入来解任何二次方程 ax² + bx + c = 0：
x = [ -b ± √b² - 4ac ] / 2a
This formula is derived by completing the square on the general quadratic equation. In IGCSE, you may be asked to use it when factorisation is difficult or when the roots are irrational.
这个公式是通过对一般二次方程配方推导出来的。在 IGCSE 中，当因式分解困难或根是无理数时，可能会要求使用这个公式。
Example: solve 2x² - 4x - 3 = 0. Here a = 2, b = -4, and c = -3. Substitute these values:
例题：解 2x² - 4x - 3 = 0。这里 a = 2，b = -4，c = -3。代入这些值：
x = [ -(-4) ± √((-4)² - 4 × 2 × (-3)) ] / (2 × 2)
x = [ 4 ± √16 + 24 ] / 4
x = [ 4 ± √40 ] / 4
x = [ 4 ± 2√10 ] / 4 = [ 2 ± √10 ] / 2
The two exact roots are (2 + √10)/2 and (2 - √10)/2. Always simplify the surd and the fraction where possible.
两个精确根为 (2 + √10)/2 和 (2 - √10)/2。在可能的情况下，一定要化简根式和分数。
6. The Discriminant and Nature of Roots | 判别式与根的性质
The expression under the square root in the quadratic formula is called the discriminant:
求根公式中平方根下的表达式称为判别式：
Δ = b² - 4ac
The discriminant tells us the nature of the roots without solving the equation. The three cases are summarised in the table below.
判别式可以在不解方程的情况下告诉我们根的性质。三种情况总结如下表。
| Discriminant | Nature of roots | 判别式 | 根的性质 |
|---|---|---|---|
| Δ > 0 | Two distinct real roots | Δ > 0 | 两个不同的实根 |
| Δ = 0 | One repeated real root | Δ = 0 | 一个重根 |
| Δ < 0 | No real roots | Δ < 0 | 没有实根 |
For example, x² - 4x + 5 = 0 has discriminant (-4)² - 4 × 1 × 5 = 16 - 20 = -4, so it has no real roots. In graph terms, the parabola does not cross the x-axis.
例如，x² - 4x + 5 = 0 的判别式为 (-4)² - 4 × 1 × 5 = 16 - 20 = -4，因此它没有实根。从图像上看，抛物线不与 x 轴相交。
IGCSE questions often ask how many real solutions a quadratic has, or require you to find the value of k for which a quadratic has equal roots. Set Δ = 0 and solve for the unknown parameter.
IGCSE 题目经常问二次方程有多少个实根，或者要求找出使方程有等根的 k 值。令 Δ = 0，然后解出未知参数。
7. Graphs of Quadratic Functions | 二次函数图像
A quadratic function y = ax² + bx + c produces a curve called a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
二次函数 y = ax² + bx + c 的图像是一条称为抛物线的曲线。如果 a > 0，抛物线开口向上，有一个最低点。如果 a < 0，抛物线开口向下，有一个最高点。
The y-intercept is found by setting x = 0, which gives y = c. The x-intercepts are the roots of ax² + bx + c = 0, if they exist. For example, y = x² - 2x - 3 crosses the y-axis at -3 and the x-axis at x = -1 and x = 3.
y 轴截距由令 x = 0 得到，即 y = c。x 轴截距是方程 ax² + bx + c = 0 的根，如果存在的话。例如，y = x² - 2x - 3 与 y 轴交于 -3，与 x 轴交于 x = -1 和 x = 3。
The axis of symmetry is the vertical line that passes through the turning point. Its equation is:
对称轴是经过顶点的竖直线。它的方程为：
x = -b / 2a
The turning point coordinates can be found by substituting this x-value into the original function, or by completing the square.
顶点坐标可以通过将这个 x 值代入原函数求得，也可以通过配方法求得。
8. Turning Point and Axis of Symmetry | 顶点与对称轴
Completing the square transforms y = ax² + bx + c into the vertex form:
配方法可以把 y = ax² + bx + c 化为顶点式：
y = a(x - h)² + k
In this form the turning point is (h, k) and the axis of symmetry is x = h. If a > 0, the turning point is a minimum; if a < 0, it is a maximum.
在这个形式中，顶点是 (h, k)，对称轴是 x = h。如果 a > 0，顶点是最低点；如果 a < 0，顶点是最高点。
Example: write y = x² - 4x + 1 in vertex form.
例题：把 y = x² - 4x + 1 写成顶点式。
y = x² - 4x + 1
y = (x² - 4x + 4) - 4 + 1
y = (x - 2)² - 3
The turning point is (2, -3) and the axis of symmetry is x = 2. Because a = 1 > 0, this parabola has a minimum value of -3.
顶点是 (2, -3)，对称轴是 x = 2。因为 a = 1 > 0，这条抛物线有最小值 -3。
9. Applications and Word Problems | 应用题与文字题
Quadratic equations model many practical situations, such as area problems, projectile motion, profit optimisation, and number puzzles. The key steps are to define a variable, form an equation, solve it, and check that the answers are sensible in context.
二次方程可以模拟许多实际问题，例如面积问题、抛体运动、利润优化和数字谜题。关键步骤是设未知数、建立方程、解方程，并根据实际情况检验答案是否合理。
Example: The product of two consecutive positive integers is 72. Find the integers.
例题：两个连续正整数的乘积是 72。求这两个整数。
Let the smaller integer be x. Then the next integer is x + 1, and their product is:
设较小的整数为 x。那么下一个整数是 x + 1，它们的乘积为：
x(x + 1) = 72
x² + x - 72 = 0
(x + 9)(x - 8) = 0
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