Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in almost every Edexcel IGCSE Mathematics (4MA1) paper, whether as direct solving questions, graph-sketching tasks, or multi-step word problems. Mastering factorisation, completing the square, the quadratic formula and the graphical method is essential for top marks.

二次方程几乎出现在每一份 Edexcel IGCSE 数学（4MA1）试卷中，无论是直接求解、函数图像题，还是多步骤应用题。熟练掌握因式分解、配方法、二次公式法和图像法，是冲击高分的关键。

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 unknown variable is 2. Its general form is ax² + bx + c = 0, where a, b and c are constants and a must not be zero.

二次方程是指未知数的最高次数为 2 的方程。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 为常数，且 a 不能为零。

If a = 0, the equation becomes linear, not quadratic.
如果 a = 0，方程就变成一次方程，而不是二次方程。
The solutions of a quadratic equation are called roots or solutions.
二次方程的解称为根（roots）。
“Quadratic” comes from the Latin word “quadratus”, meaning square.
“Quadratic” 一词源自拉丁文 “quadratus”，意为”平方”。

2. Rearranging into Standard Form | 化为标准形式

Before choosing a solving method, you must always rearrange the equation so that one side equals zero and the terms are in descending powers of x.

在选择解法之前，你必须先把方程整理成一边等于零，并且各项按 x 的降幂排列。

Example: Solve 3x(x − 1) = 2x + 5.

例：解方程 3x(x − 1) = 2x + 5。

Expand the left-hand side first:
先展开左边：

3x² − 3x = 2x + 5 → 3x² − 5x − 5 = 0

Expand brackets carefully, watching negative signs.
展开括号时小心负号。
Collect all terms on one side; the x² term should be positive if possible.
把所有项移到一边；x² 项的系数最好为正。
If the equation contains fractions, multiply through by the common denominator first.
若方程含分数，先乘以公分母去分母。

3. Solving by Factorisation | 因式分解法

Factorisation works when the quadratic can be written as a product of two linear brackets. This is often the fastest method in the IGCSE exam.

当二次式能写成两个一次括号相乘时，就可以用因式分解法。这通常是 IGCSE 考试中最快的方法。

Worked example: Solve x² − 5x + 6 = 0.
例：解 x² − 5x + 6 = 0。

Find two numbers that multiply to +6 and add to −5. The numbers are −2 and −3, so:
找两个数，相乘等于 +6，相加等于 −5。这两个数是 −2 和 −3，所以：

(x − 2)(x − 3) = 0 → x = 2 or x = 3

Always set each bracket equal to zero separately.
务必让每个括号分别等于零。
Check your roots by substituting them back into the original equation.
把根代回原方程检验。
Special case: x² − 9 = 0 → (x − 3)(x + 3) = 0 → x = ±3 (difference of two squares).
特殊情况：x² − 9 = 0 → (x − 3)(x + 3) = 0 → x = ±3（平方差公式）。
Special case: 2x² + 4x = 0 → 2x(x + 2) = 0 → x = 0 or x = −2 (common factor).
特殊情况：2x² + 4x = 0 → 2x(x + 2) = 0 → x = 0 或 x = −2（提取公因式）。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. It is useful when factorisation is not obvious and it also reveals the turning point of the graph.

配方法把二次式改写成 (x + p)² + q 的形式。当因式分解不明显时非常有用，同时还能直接看出图像的顶点。

Worked example: Solve x² + 6x + 2 = 0.
例：解 x² + 6x + 2 = 0。

Take half of 6, which is 3, and write:
取 6 的一半，即 3，写出：

(x + 3)² − 9 + 2 = 0 → (x + 3)² = 7

x + 3 = ±√7 → x = −3 ± √7

Remember to subtract the square of half the coefficient: (b/2)².
不要忘记减去”一次项系数一半的平方”：(b/2)²。
The vertex of y = (x + 3)² − 7 is at (−3, −7).
函数 y = (x + 3)² − 7 的顶点坐标为 (−3, −7)。
Leave answers in surd form unless the question asks for decimals.
除非题目要求小数，否则答案保留根号形式。

5. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation, even when it cannot be factorised easily. You must memorise it for the exam.

二次公式可以求解任何二次方程，即使无法轻松因式分解。考试中你必须熟记这个公式。

x = (−b ± √(b² − 4ac)) / (2a)

Here a, b and c are taken from the standard form ax² + bx + c = 0. Substitute carefully, paying attention to signs.

这里的 a、b、c 取自标准形式 ax² + bx + c = 0。代入时要格外小心符号。

Write down a =, b =, c = before substituting.
代入前先写出 a =、b =、c =。
Use a scientific calculator to check your final decimals.
用科学计算器检验最终小数答案。
Do not round until the last step.
最后一步之前不要四舍五入。

6. Worked Example Using the Formula | 公式法例题

Solve 2x² − 4x − 3 = 0 using the quadratic formula.
用二次公式解 2x² − 4x − 3 = 0。

Here a = 2, b = −4 and c = −3. Substitute these values:
这里 a = 2，b = −4，c = −3。代入公式：

x = (4 ± √((−4)² − 4 × 2 × (−3))) / (2 × 2)

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

Simplify √40 = 2√10, so x = (2 ± √10) / 2. As decimals, x ≈ 2.58 or x ≈ −0.58.

化简 √40 = 2√10，所以 x = (2 ± √10) / 2。化为小数得 x ≈ 2.58 或 x ≈ −0.58。

A negative sign inside the square root means you have made a sign error.
根号内若出现负数，说明你符号代错了。
Check that b² − 4ac is simplified before taking the square root.
开根号之前，先化简 b² − 4ac。
Give both roots; the ± symbol represents two distinct answers.
两个根都要写出；± 代表两个不同的答案。

7. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells you how many real roots a quadratic equation has without solving it.

式子 b² − 4ac 称为判别式。不解方程，它就能告诉你二次方程有几个实数根。

Discriminant 判别式	Number of real roots 实数根个数	Graph meaning 图像含义
b² − 4ac > 0	Two distinct real roots 两个不同实数根	Graph crosses the x-axis twice 图像与 x 轴有两个交点
b² − 4ac = 0	One repeated real root 两个相等实数根	Graph touches the x-axis at one point 图像与 x 轴相切于一点
b² − 4ac < 0	No real roots 无实数根	Graph does not meet the x-axis 图像与 x 轴无交点

If b² − 4ac is a perfect square, the roots are rational and factorisation should work. If not, the roots are irrational surds.

若 b² − 4ac 是完全平方数，则根为有理数，通常可以因式分解；若不是，则根为无理数（根式形式）。

8. Solving Quadratic Equations Using Graphs | 用图像法解二次方程

The roots of ax² + bx + c = 0 are exactly the x-intercepts of the graph y = ax² + bx + c. Reading where the curve crosses the x-axis gives approximate solutions.

方程 ax² + bx + c = 0 的根，正是图像 y = ax² + bx + c 与 x 轴交点的横坐标。读出曲线与 x 轴的交点即可得到近似解。

Example: The graph of y = x² − 3x + 2 crosses the x-axis at x = 1 and x = 2, so the equation x² − 3x + 2 = 0 has roots x = 1 and x = 2.

例：y = x² − 3x + 2 的图像与 x 轴交于 x = 1 和 x = 2，因此方程 x² − 3x + 2 = 0 的根为 x = 1 和 x = 2。

If the parabola does not touch the x-axis, the equation has no real roots.
若抛物线与 x 轴没有交点，则方程无实数根。
The completed-square form y = (x − h)² + k gives the turning point (h, k).
配方形式 y = (x − h)² + k 可直接给出顶点坐标 (h, k)。
To solve ax² + bx + c = k graphically, draw the line y = k and read the intersection x-values.
要在图像上解 ax² + bx + c = k，可画水平线 y = k，读取交点横坐标。

9. Word Problems Involving Quadratics | 二次方程应用题

Exam word problems often lead to a quadratic equation after forming and simplifying an area or product expression.

考试中的应用题往往在列出并化简面积或乘积表达式后，得到二次方程。

Worked example: A rectangle has length (x + 4) m and width (x + 1) m. Its area is 40 m². Find x.

例：一个长方形的长为 (x + 4) 米，宽为 (x + 1) 米，面积为 40 平方米。求 x。

(x + 4)(x + 1) = 40 → x² + 5x + 4 = 40

x² + 5x − 36 = 0 → (x + 9)(x − 4) = 0

So x = −9 or x = 4. Since lengths cannot be negative, reject x = −9. The valid answer is x = 4.

所以 x = −9 或 x = 4。由于长度不能为负，舍去 x = −9，有效答案为 x = 4。

Always state why you reject a negative root in a context question.
在应用题中，务必说明为什么舍去负根。
Use the original units of measurement in your final answer.
最终答案使用题目原有的单位。
Read the question again: sometimes x is only part of what is asked.
再读一遍题目：有时 x 只是所求量的一部分。

10. Common Exam Mistakes | 常见考试误区

Here are the most frequent errors students make in quadratic questions, with advice on how to avoid them.

下面是学生在二次方程题中最常犯的错误，以及避免方法。

Forgetting to rearrange: solving 2x² + 5 = 3x without moving all terms to one side.
忘记整理：解 2x² + 5 = 3x 时没有把所有项移到一边。
Dividing by x: this loses the root x = 0. Factor instead of dividing.
两边除以 x：这会丢失根 x = 0。应该提取公因式，而不是约去 x。
Misreading signs in the formula: when b is negative, −b becomes positive.
公式中符号看错：当 b 为负数时，−b 要变为正数。
Using the quadratic formula without writing a, b and c first.
不先写出 a、b、c 就直接套用二次公式。
Rounding intermediate values, which destroys accuracy.
过早四舍五入中间值，导致精度损失。

11. Exam Checklist | 考场检查清单

Use this quick checklist before you move on to the next question.

进入下一题之前，快速对照这份检查清单。

Check 检查项	Question to ask yourself 自问
Standard form 标准形式	Is the equation written as ax² + bx + c = 0? 方程是否写成 ax² + bx + c = 0？
Method 解法	Is factorisation possible, or should I use the formula? 可否因式分解？否则用公式法？
Discriminant 判别式	Is b² − 4ac positive, zero or negative? b² − 4ac 是正、零还是负？
Substitution 代入检验	Do both roots satisfy the original equation? 两个根是否都满足原方程？
Context 情境	Are both roots sensible in the problem context? 在题目情境中两个根是否都合理？

12. Summary | 总结

Quadratic equations are a guaranteed topic on the Edexcel IGCSE Mathematics exam. Practise rearranging into standard form, choose the fastest method, check the discriminant and verify every root by substitution.

二次方程是 Edexcel IGCSE

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Solving Quadratic Equations | 解二次方程