📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程
Quadratic equations are one of the most important and frequently examined topics in IGCSE Mathematics. They appear in algebra questions, graph sketching, and real-life word problems. A clear understanding of quadratic equations can help you earn valuable marks in both Core and Extended papers.
二次方程是 IGCSE 数学中最重要、最常考的主题之一。它们出现在代数题、图像绘制题和实际生活应用题中。清楚地理解二次方程可以帮助你在核心和扩展试卷中获得宝贵分数。
1. What Is a Quadratic Equation? | 什么是二次方程
are constants and a ≠ 0. The variable is usually written as x, but any letter can be used.
二次方程是未知数的最高次数为 2 的方程。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。变量通常写成 x，但也可以使用任何字母。
er of x is 2. In contrast, 4x − 7 = 0 is linear, because the highest power of x is only 1.
例如，2x² + 3x − 5 = 0 是一个二次方程，因为 x 的最高次数是 2。相比之下，4x − 7 = 0 是一次方程，因为 x 的最高次数仅为 1。
Quadratic equations usually have two solutions, although the two solutions can sometimes be equal or not real. These solutions are called the roots of the equation.
二次方程通常有两个解，尽管这两个解有时可能相等或不是实数。这些解称为方程的根。
2. Standard Form and Key Terms | 标准形式与关键术语
standard form is ax² + bx + c = 0, where the terms are ordered by decreasing powers of x.
在解二次方程之前，通常需要将其写成标准形式。标准形式为 ax² + bx + c = 0，其中各项按照 x 的降幂排列。
is not in standard form. Rearranging gives 3x² − 5x + 2 = 0. Here a = 3, b = −5 and c = 2.
例如，方程 3x² = 5x − 2 不是标准形式。移项后得到 3x² − 5x + 2 = 0。此时 a = 3，b = −5，c = 2。
ax² + bx + c = 0, a ≠ 0
- a is called the leading coefficient of x².
- b is the coefficient of x.
- c is the constant term.
- a 称为 x² 的二次项系数。
- b 是 x 的一次项系数。
- c 是常数项。
Always make sure the right-hand side is 0 before factorising or applying the quadratic formula. This is a simple but very important habit.
在进行因式分解或使用二次公式之前，务必确保方程右边为 0。这是一个简单但非常重要的习惯。
3. Solving by Factorisation | 因式分解法求解
Factorisation is often the fastest way to solve a quadratic equation when the factors are easy to find. The method works because if the product of two expressions is zero, then at least one of the expressions must be zero.
当因式容易找到时，因式分解通常是解二次方程最快的方法。该方法之所以有效，是因为如果两个式子的乘积为零，那么其中至少有一个式子必须为零。
k for two numbers that multiply to 6 and add to 5. The numbers are 2 and 3, so we can write:
例如，解 x² + 5x + 6 = 0。我们寻找两个数，它们的乘积为 6，和为 5。这两个数是 2 和 3，因此可以写成：
x² + 5x + 6 = (x + 2)(x + 3) = 0
x + 2 = 0 or x + 3 = 0. This gives x = −2 or x = −3. The two roots are x = −2 and x = −3.
然后 x + 2 = 0 或 x + 3 = 0。由此得到 x = −2 或 x = −3。两个根为 x = −2 和 x = −3。
Factorisation becomes more challenging when a is not equal to 1. In that case, you may need to split the middle term or use trial and error. For instance, 2x² + 7x + 3 can be factorised as (2x + 1)(x + 3).
当 a 不等于 1 时，因式分解会变得更有挑战性。这时你可能需要拆分中项或进行试错。例如，2x² + 7x + 3 可以因式分解为 (2x + 1)(x + 3)。
4. Difference of Two Squares | 平方差公式
he identity is a² − b² = (a + b)(a − b). This is extremely useful when there is no x term.
二次因式分解的一种特殊情况是平方差。其恒等式为 a² − b² = (a + b)(a − b)。当方程中没有 x 项时，这个公式非常有用。
For example, solve x² − 9 = 0. This can be written as x² − 3² = 0, so:
例如，解 x² − 9 = 0。它可以写成 x² − 3² = 0，因此：
x² − 9 = (x + 3)(x − 3) = 0
The solutions are x = −3 or x = 3. Notice that one root is the negative of the other.
解为 x = −3 或 x = 3。注意其中一个根是另一个根的相反数。
larly, 4x² − 25 = 0 can be factorised as (2x + 5)(2x − 5) = 0, giving x = −5/2 or x = 5/2.
类似地，4x² − 25 = 0 可以因式分解为 (2x + 5)(2x − 5) = 0，得到 x = −5/2 或 x = 5/2。
Recognising the difference of two squares can save a lot of time in the exam. Look carefully for expressions such as x² − 16, 9x² − 1, or 25 − 4x².
识别平方差可以在考试中节省大量时间。仔细寻找诸如 x² − 16、9x² − 1 或 25 − 4x² 这样的式子。
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. It is especially useful for finding the vertex of a parabola and for solving equations that do not factorise easily.
配方法是一种将二次表达式改写为 (x + p)² + q 形式的方法。它对于求抛物线的顶点以及解不容易因式分解的方程特别有用。
leting the square. First, take the coefficient of x, which is 6, halve it to get 3, and square it to get 9:
例如，用配方法解 x² + 6x + 2 = 0。首先，取 x 的系数 6，除以 2 得到 3，再平方得到 9：
x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7
0. Adding 7 to both sides gives (x + 3)² = 7. Taking the square root of both sides gives:
于是方程变为 (x + 3)² − 7 = 0。两边加 7 得到 (x + 3)² = 7。两边开平方得到：
x + 3 = ±√7
Therefore x = −3 ± √7. The exact roots are x = −3 + √7 and x = −3 − √7.
因此 x = −3 ± √7。精确根为 x = −3 + √7 和 x = −3 − √7。
When a is not 1, factor out the leading coefficient first. For example, 2x² + 8x + 5 should be written as 2(x² + 4x) + 5 before completing the square inside the bracket.
当 a 不等于 1 时，首先提取二次项系数。例如，2x² + 8x + 5 应写成 2(x² + 4x) + 5，然后在括号内完成配方。
6. The Quadratic Formula | 二次公式
impossible. For an equation in standard form ax² + bx + c = 0, the solutions are given by:
二次公式可以解任何二次方程，即使因式分解很困难或不可能。对于标准形式 ax² + bx + c = 0 的方程，其解为：
x = (−b ± √b² − 4ac) ÷ 2a
2x² + 4x − 3 = 0. Here a = 2, b = 4 and c = −3. Substitute these values into the formula:
例如，解 2x² + 4x − 3 = 0。这里 a = 2，b = 4，c = −3。将这些值代入公式：
x = (−4 ± √(4² − 4 × 2 × (−3))) ÷ (2 × 2)
x = (−4 ± √16 + 24) ÷ 4
x = (−4 ± √40) ÷ 4 = −1 ± √10 ÷ 2
The exact roots are x = −1 + √10/2 and x = −1 − √10/2. The final answer should be simplified as much as possible.
精确根为 x = −1 + √10/2 和 x = −1 − √10/2。最终答案应尽可能化简。
Remember to write the formula carefully and show every step. A small sign error in b or c can change the entire answer.
记住要仔细写出公式并展现每一步。b 或 c 的一个小符号错误就可能改变整个答案。
7. The Discriminant | 判别式
The discriminant is the part of the quadratic formula under the square root sign: b² − 4ac. It tells us how many real roots a quadratic equation has without actually solving it.
判别式是二次公式中平方根号下的部分：b² − 4ac。它可以在不求根的情况下告诉我们二次方程有多少个实数根。
| Discriminant value | Number of real roots | Meaning |
| b² − 4ac > 0 | 2 distinct real roots | The graph cuts the x-axis twice. |
| b² − 4ac = 0 | 1 repeated real root | The graph touches the x-axis once. |
| b² − 4ac < 0 | no real roots | The graph does not meet the x-axis. |
it has one repeated root. In fact, x² − 4x + 4 = (x − 2)², so the repeated root is x = 2.
例如，x² − 4x + 4 = 0 的判别式为 (−4)² − 4 × 1 × 4 = 16 − 16 = 0，因此它有一个重根。事实上，x² − 4x + 4 = (x − 2)²，所以重根为 x = 2。
Discriminant questions often ask whether a quadratic has two distinct roots, one repeated root, or no real roots. This is a quick, high-mark topic if you memorise the rule.
判别式题目常常问一个二次方程有两个不同根、一个重根还是没有实数根。如果你记住了规则，这是一个可以快速得分的考点。
8. Graphical Interpretation | 图像解释
= ax² + bx + c has a U-shape if a is positive and an upside-down U-shape if a is negative.
二次方程对应的图像是一条称为抛物线的曲线。y = ax² + bx + c 的图像在 a 为正时呈 U 形，在 a 为负时呈倒 U 形。
re the parabola crosses the x-axis. If the parabola does not cross the x-axis, the equation has no real roots.
二次方程 ax² + bx + c = 0 的根是抛物线与 x 轴交点的横坐标。如果抛物线不与 x 轴相交，则该方程没有实数根。
iven by x = −b ÷ 2a. The axis of symmetry is the vertical line passing through the vertex.
抛物线的顶点是其最高点或最低点。顶点的横坐标为 x = −b ÷ 2a。对称轴是经过顶点的竖直线。
(x − 3) = 0. Its vertex has x-coordinate −(−4) ÷ 2 = 2, and the axis of symmetry is x = 2.
例如，y = x² − 4x + 3 在 x = 1 和 x = 3 处有根，因为 (x − 1)(x − 3) = 0。其顶点的横坐标为 −(−4) ÷ 2 = 2，对称轴为 x = 2。
9. Quadratic Word Problems | 二次方程应用题
Quadratic equations often model real-life situations involving area, product of numbers, or projectile motion. The hardest part is usually translating the words into an equation.
二次方程常用于建立涉及面积、数的乘积或抛体运动等实际问题的模型。最困难的部分通常是把文字转化为方程。
For example, the product of two consecutive positive integers is 132. Let the smaller integer be x, so the larger is x + 1. The equation is:
例如，两个连续正整数的乘积为 132。设较小的整数为 x，则较大的为 x + 1。方程为：
x(x + 1) = 132
x² + x − 132 = 0
= 11 or x = −12. Since the problem asks for positive integers, the solution is 11 and 12.
因式分解得到 (x + 12)(x − 11) = 0，因此 x = 11 或 x = −12。由于题目要求正整数，所以答案是 11 和 12。
x = 5 or x = −8. Since length cannot be negative, the width is 5 m and the length is 8 m.
另一种常见类型是面积问题。一个长方形花园的宽为 x 米，长为 (x + 3) 米。若面积为 40 平方米，则 x(x + 3) = 40，即 x² + 3x − 40 = 0。解得 x = 5 或 x = −8。由于长度不能为负，宽为 5 米，长为 8 米。
Always check that your mathematical answers make sense in the original context. Reject negative or unrealistic solutions when the problem requires positive quantities.
始终检查你的数学答案在原始情境中是否合理。当题目要求正数时，应舍去负数或不切实际的解。
10. Choosing the Right Method | 选择合适的方法
There are several ways to solve a quadratic equation, and choosing the right method can save time. Use factorisation when the factors are obvious. Use the quadratic formula when the equation does not factorise easily. Use completing the square when you need the exact form of the vertex.
解二次方程有几种方法，选择合适的方法可以节省时间。当因式显而易见时使用因式分解。当方程不容易因式分解时使用二次公式。当需要顶点的精确形式时使用配方法。
| Method | Best used when |
| Factorisation | Factors are simple and easy to find. |
| Quadratic formula | Any equation, especially awkward numbers. |
| Completing the square | Finding vertex, exact form, or proving results. |
| Graphically | Approximate solutions or interpreting roots. |
In the exam, if the question says "give your answer correct to 2 decimal places", this often indicates the quadratic formula is needed. If the question says "factorise fully", you should use factorisation.
在考试中，如果题目说“答案精确到 2 位小数”，这通常提示需要使用二次公式。如果题目说“完全因式分解”，则应使用因式分解。
Whichever method you choose, always write down the standard form first. This reduces mistakes and makes your working clear to the examiner.
无论选择哪种方法，都应先写出标准形式。这样可以减少错误，并使你的解题过程对阅卷老师清晰明了。
11. Common Mistakes to Avoid | 常见错误与避免方法
ample, if you have x² + 3x = 4, you must rewrite it as x² + 3x − 4 = 0 before factorising.
最常见的错误之一是在将所有项移到同一边之前就尝试因式分解。例如，如果方程是 x² + 3x = 4，必须先改写成 x² + 3x − 4 = 0，然后再因式分解。
e root x = 0. The correct method is x² − 5x = 0, then x(x − 5) = 0, giving x = 0 or x = 5.
另一个常见错误是两边同时除以 x。例如，从 x² = 5x 出发，一些学生除以 x 得到 x = 5，但他们丢失了根 x = 0。正确的方法是 x² − 5x = 0，然后 x(x − 5) = 0，得到 x = 0 或 x = 5。
- Forgetting the ± sign in the quadratic formula when taking square roots.
- Incorrectly identifying a, b or c, especially when the equation is not in standard form.
- Arithmetic mistakes inside the square root when calculating the discriminant.
- Leaving a negative solution when the context requires a positive value.
- 在使用二次公式开平方时忘记 ± 号。
- 当方程不是标准形式时，错误地确定 a、b 或 c。
- 计算判别式时在平方根内出现算术错误。
- 当题意要求正值时，仍保留负数解。
Careful line-by-line working and a quick substitution of your final roots back into the original equation can catch most of these errors.
逐行仔细计算，并快速将最终根代回原方程检验，可以发现大多数此类错误。
12. Summary and Exam Tips | 总结与考试技巧
Quadratic equations are a central part of IGCSE Mathematics. You should be confident in writing an equation in standard form, solving it by factorisation, completing the square and the quadratic formula, and using the discriminant to interpret the roots.
二次方程是 IGCSE 数学的核心内容。你应当熟练掌握将方程写成标准形式、用因式分解法、配方法和二次公式求解，以及使用判别式判断根的情况。
Before an exam, practise mixed questions so you can decide quickly which method is best. Show all working because method marks are often awarded even if the final answer is wrong.
考试前，练习混合题型，以便快速决定哪种方法最合适。展示所有解题过程，因为即使最终答案错误，过程分也常常可以获得。
- Always set the equation to 0 before solving.
- Factorise if possible; otherwise use the quadratic formula.
- Use the discriminant to check the number of roots.
- Substitute your roots back into the original equation to verify them.
- Leave answers in exact form unless the question asks for decimals.
- 解方程前始终将方程设为 0。
- 如果可以因式分解就先因式分解；否则使用二次公式。
- 使用判别式检查根的个数。
- 将根代回原方程进行验证。
- 除非题目要求小数，否则保留精确形式。
With regular practice, quadratic equations can become one of the most reliable sources of marks in your IGCSE Mathematics paper.
通过定期练习，二次方程可以成为你 IGCSE 数学试卷中最可靠的得分来源之一。
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