📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程
Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, graph sketching, problem solving and even in geometry and mensuration questions. A strong understanding of quadratic equations will help you gain marks across both Core and Extended papers.
二次方程是 IGCSE 数学课程中最重要的主题之一。它们出现在代数、图像绘制、问题解决，甚至几何和求积问题中。扎实掌握二次方程将帮助你在核心和扩展试卷中获得更多分数。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation in which the highest power of the variable is 2. It can contain an x² term, an x term and a constant term, but it cannot contain a higher power such as x³. The graph of a quadratic expression is always a parabola.
二次方程是指变量最高次数为 2 的方程。它可以包含 x² 项、x 项和常数项，但不能包含 x³ 等更高次项。二次表达式的图像总是抛物线。
The most common general form of a quadratic equation is shown below. The coefficient a must not be zero because otherwise the equation would become linear.
二次方程最常见的一般形式如下所示。系数 a 不能为零，否则方程将变为一次方程。
ax² + bx + c = 0, a ≠ 0
- Example: x² - 5x + 6 = 0
示例：x² - 5x + 6 = 0 - Example: 3x² + 2x - 1 = 0
示例：3x² + 2x - 1 = 0
2. Standard Form and Key Vocabulary | 标准形式与关键术语
ide equals zero, and the terms are usually arranged with the x² term first, then the x term, then the constant.
在求解二次方程之前，你应始终将其写成标准形式：ax² + bx + c = 0。这意味着方程一边等于零，项通常按 x² 项、x 项、常数项的顺序排列。
2 makes the equation true, then 2 is a root or solution. A quadratic equation can have two roots, one repeated root or no real roots.
数值 a、b 和 c 称为系数。方程的解也称为根。例如，如果 x = 2 使方程成立，那么 2 就是一个根或解。二次方程可以有两个根、一个重根或没有实根。
| Term 术语 | Meaning 含义 |
| Coefficient | The number multiplying a variable, such as a, b and c in ax² + bx + c |
| Root / Solution | A value of x that makes the equation true |
| Parabola | The U-shaped graph of a quadratic function |
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic expression factorises neatly. First write the equation in standard form, then factorise the left-hand side into two brackets, and finally set each bracket equal to zero.
当二次表达式可以整齐地因式分解时，因式分解法通常是最快的方法。首先将方程写成标准形式，然后将左边分解为两个括号，最后令每个括号等于零。
and add to -7. These numbers are -3 and -4, so the factorised form is (x - 3)(x - 4) = 0.
例如，要求解 x² - 7x + 12 = 0，找到两个数，它们的乘积为 12，和为 -7。这两个数是 -3 和 -4，因此因式分解形式为 (x - 3)(x - 4) = 0。
(x - 3)(x - 4) = 0
x = 3 or x = 4
- Always check that the equation equals zero before factorising.
- 先检查方程是否等于零，再进行因式分解。
- Difference of two squares: x² - 9 = (x - 3)(x + 3).
- 平方差：x² - 9 = (x - 3)(x + 3)。
- Do not divide both sides by x, because you may lose the solution x = 0.
- 不要两边同时除以 x，因为这样可能丢失 x = 0 这个解。
4. The Quadratic Formula | 求根公式法
The quadratic formula works for every quadratic equation, even when factorising is difficult or impossible with simple integers. You should memorise the formula exactly and substitute the values of a, b and c carefully.
求根公式适用于每一个二次方程，即使因式分解很困难或无法用简单整数完成。你应准确记住该公式，并仔细代入 a、b 和 c 的值。
x = (−b ± √b² − 4ac) / (2a)
= -2. Substitute these values into the formula and simplify the square root step by step.
对于方程 2x² + 3x - 2 = 0，确定 a = 2，b = 3，c = -2。将这些值代入公式，并逐步化简平方根。
x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2)
x = (−3 ± √9 + 16) / 4
x = (−3 ± 5) / 4
ula is especially useful in Extended questions where decimal or surd answers are required.
这给出两个解 x = 1/2 和 x = -2。在扩展题中，当需要小数或根式答案时，求根公式尤其有用。
5. Completing the Square | 配方法
Completing the square is a method that rewrites a quadratic expression in the form (x + p)² + q. This form is very useful for finding the vertex of a parabola, solving equations and proving results.
配方法是将二次表达式改写为 (x + p)² + q 的形式。这种形式对于求抛物线的顶点、求解方程和证明结论非常有用。
: x² + 8x = -5. Then add (8/2)² = 16 to both sides to make the left side a perfect square.
要求解 x² + 8x + 5 = 0，先将常数移到右边：x² + 8x = -5。然后两边加上 (8/2)² = 16，使左边成为完全平方。
x² + 8x + 16 = 11
(x + 4)² = 11
x + 4 = ±√11
x = −4 ± √11
Remember that when the coefficient of x² is not 1, you must divide all terms by that coefficient before completing the square.
请记住，当 x² 的系数不是 1 时，必须先将所有项除以该系数，再进行配方。
6. The Discriminant and Number of Roots | 判别式与根的个数
- 4ac. It tells you how many real roots a quadratic equation has without fully solving it.
判别式是求根公式中平方根号下的部分：Δ = b² - 4ac。它可以在不完全求解的情况下告诉你二次方程有多少个实根。
Δ = b² − 4ac
| Condition 条件 | Number of real roots 实根个数 |
| Δ > 0 | Two distinct real roots 两个不同实根 |
| Δ = 0 | One repeated real root 一个重根 |
| Δ < 0 | No real roots 没有实根 |
eated root x = 2. The equation x² + x + 1 = 0 has Δ = 1 - 4 = -3, so it has no real roots.
例如，x² - 4x + 4 = 0 的判别式 Δ = (−4)² - 4(1)(4) = 0，因此它有一个重根 x = 2。方程 x² + x + 1 = 0 的 Δ = 1 - 4 = -3，因此没有实根。
7. Graphs of Quadratic Functions | 二次函数图像
. If a is positive, the parabola opens upward with a minimum point. If a is negative, it opens downward with a maximum point.
y = ax² + bx + c 的图像是抛物线。如果 a 为正，抛物线开口向上，有最低点。如果 a 为负，抛物线开口向下，有最高点。
dratic equation ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.
对称轴是竖直的，并穿过顶点。其方程为 x = −b / (2a)。二次方程 ax² + bx + c = 0 的根是图像与 x 轴交点的横坐标。
Vertex x-coordinate: x = −b / (2a)
- If Δ > 0, the graph crosses the x-axis twice.
- 如果 Δ > 0，图像与 x 轴相交两次。
- If Δ = 0, the graph touches the x-axis at exactly one point.
- 如果 Δ = 0，图像恰好在一个点接触 x 轴。
- If Δ < 0, the graph does not intersect the x-axis at all.
- 如果 Δ < 0，图像完全不与 x 轴相交。
8. Using Quadratic Equations in Word Problems | 二次方程应用题
Many IGCSE word problems lead to quadratic equations. You should define a variable, form an equation, solve it, and then check whether the answer makes sense in the original context. Negative roots are often rejected in measurement questions.
许多 IGCSE 应用题会列出二次方程。你应定义变量，建立方程，求解，然后检查答案在原情境中是否合理。在测量问题中，负根通常会被舍去。
For example, a rectangle has a width of x metres and a length that is 3 metres more than its width. If the area is 40 m², form an equation and solve it.
例如，一个长方形的宽为 x 米，长比宽多 3 米。如果面积为 40 m²，建立方程并求解。
x(x + 3) = 40
x² + 3x − 40 = 0
(x + 8)(x − 5) = 0
x = 5 or x = −8
Since width cannot be negative, the width is 5 metres and the length is 8 metres. Always state the units and check that both dimensions satisfy the original area condition.
由于宽度不能为负数，因此宽为 5 米，长为 8 米。始终标明单位，并检查两个尺寸是否满足原始面积条件。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
other mistake is dividing both sides of an equation by x, which can delete the root x = 0.
一个常见错误是在确定 a、b 和 c 之前忘记将方程写成标准形式。另一个错误是将方程两边同时除以 x，这可能会删去根 x = 0。
gative, the formula gives −b, which becomes positive. For example, if b = -3, then −b = 3.
学生也经常在求根公式中混淆 b 的符号。如果 b 为负，公式中的 −b 将变为正。例如，如果 b = -3，那么 −b = 3。
- Always expand and simplify before factorising.
- 因式分解前务必展开并化简。
- Do not divide by x unless you know x is not zero.
- 除非你知道 x 不为零，否则不要除以 x。
- Check your roots by substituting them back into the original equation.
- 将根代回原方程进行检验。
- In word problems, reject answers that do not fit the real-world context.
- 在应用题中，舍去不符合现实情境的答案。
10. Exam Strategy and Practice Tips | 考试策略与练习建议
In the IGCSE exam, read the question carefully and decide whether factorising, the quadratic formula or completing the square is quickest. If the question asks for answers correct to a certain number of decimal places, the quadratic formula is usually the best choice.
在 IGCSE 考试中，仔细阅读题目，判断用因式分解、求根公式还是配方法最快。如果题目要求答案精确到一定小数位，通常求根公式是最佳选择。
Show every line of working and write the formula before substituting values. This helps you gain method marks even if a small arithmetic error occurs. Practise past paper questions regularly and time yourself to build speed and accuracy.
展示每一行解题过程，并在代入数值前写出公式。这样即使出现小的计算错误，也能帮助你获得方法分。定期练习历年真题并计时，以提高速度和准确性。
- Memorise the quadratic formula and discriminant condition.
- 熟记求根公式和判别式条件。
- Practise completing the square with both positive and negative coefficients.
- 练习正、负系数下的配方法。
- Sketch the graph to check the number of roots visually.
- 通过画图直观检查根的个数。
- Write final answers clearly, including units where necessary.
- 清晰写出最终答案，必要时包括单位。
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