Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. They appear in factorisation, coordinate geometry, word problems and even in the extended paper’s final questions. Mastering the three main solution strategies — factorisation, completing the square and the quadratic formula — will give you a reliable toolkit for any question the examiner throws at you.

二次方程是IGCSE数学中考查频率最高的内容之一。它出现在因式分解、坐标几何、应用题，甚至扩展卷的最后几道大题中。掌握三种主要解法——因式分解法、配方法和求根公式——能让你在考场上应对自如。

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

A quadratic equation is a polynomial equation of degree 2. Its general form is:

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are real numbers, and a cannot be zero — otherwise the equation becomes linear. The expression ax² + bx + c is called the quadratic expression, and the highest power of x is always 2.

二次方程是次数为2的多项式方程，其一般形式为：

ax² + bx + c = 0，其中 a ≠ 0

这里 a、b、c 是实数，且 a 不能为0——否则方程就变成了一次方程。ax² + bx + c 被称为二次表达式，x 的最高次数始终是2。

• If b = 0, the equation becomes ax² + c = 0. For example, x² − 9 = 0 can be solved by difference of squares.
• 如果 b = 0，方程变为 ax² + c = 0。例如 x² − 9 = 0 可用平方差公式求解。
• If c = 0, the equation becomes ax² + bx = 0. For example, x² − 5x = 0 can be factorised by taking out a common factor.
• 如果 c = 0，方程变为 ax² + bx = 0。例如 x² − 5x = 0 可通过提取公因式来分解。

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2. The Zero-Product Property and Factorisation | 零积性质与因式分解法

The simplest way to solve a quadratic equation is by factorisation, supported by the zero-product property:

If A × B = 0, then A = 0 or B = 0.

This property is true because the product of two non-zero numbers is never zero. So once we factor the quadratic into two brackets, we can solve two linear equations.

最简单的方法是使用因式分解法，其依据是零积性质：

若 A × B = 0，则 A = 0 或 B = 0。

这个性质成立是因为两个非零数的乘积不可能为0。因此，只要把二次式分解成两个括号的乘积，就可以分别解两个一次方程。

Worked example | 例题： Solve x² − 7x + 12 = 0.

We look for two numbers that multiply to give 12 and add to give −7. The numbers are −3 and −4:

我们要找两个数，它们相乘得12，相加得−7。这两个数是 −3 和 −4：

(x − 3)(x − 4) = 0

x − 3 = 0 或 x − 4 = 0

x = 3 或 x = 4

• Always arrange the equation in the form ax² + bx + c = 0 before factorising. Many students forget to move all terms to one side first.
• 在分解因式之前，一定要先把方程整理成 ax² + bx + c = 0 的形式。很多同学忘记先把所有项移到一边。
• If the equation cannot be factorised easily, use completing the square or the quadratic formula instead.
• 如果方程不易分解，就改用配方法或求根公式。

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3. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form (x + p)² + q. The key identity is:

配方法将二次表达式改写成 (x + p)² + q 的形式。关键恒等式是：

x² + bx = (x + b/2)² − (b/2)²

In words: halve the coefficient of x, square it, and subtract that square. This technique is especially useful for solving equations and for finding the turning point of a parabola.

简单来说：把 x 的一次项系数除以2，再平方，然后减去这个平方。这一技巧在解方程和求抛物线顶点时特别有用。

Worked example | 例题： Solve x² + 6x + 5 = 0 by completing the square.

Step 1: Move the constant to the right-hand side, or keep it and compensate. Here we isolate the x² + 6x part:

第一步：把常数项移到右边，或者保留并补正。这里我们先处理 x² + 6x 部分：

x² + 6x = −5

Step 2: Half of 6 is 3, and 3² = 9. Add 9 to both sides:

第二步：6 的一半是3，而 3² = 9。两边同时加9：

x² + 6x + 9 = 4

(x + 3)² = 4

Step 3: Take the square root of both sides, remembering the ± sign:

第三步：两边开平方，注意±号：

x + 3 = ±2

x = −3 ± 2, so x = −1 或 x = −5

When the coefficient of x² is not 1, first factor it out. For example, 2x² + 8x + 3 = 0 becomes 2(x² + 4x) + 3 = 0, then complete the square inside the bracket.

当 x² 的系数不是1时，先把它提取出来。例如 2x² + 8x + 3 = 0 先化为 2(x² + 4x) + 3 = 0，再在括号内配方。

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4. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation in one step. It is derived by completing the square on ax² + bx + c = 0:

求根公式可以一步解出任何二次方程。它通过对 ax² + bx + c = 0 配方推导而来：

x = (−b ± √(b² − 4ac)) / 2a

To use it, identify a, b and c from the equation, substitute them carefully, and simplify. Take special care with negative signs.

使用时，先确定方程中的 a、b、c，然后仔细代入并化简。尤其要注意负号。

Worked example | 例题： Solve 2x² + 3x − 5 = 0.

Here a = 2, b = 3, c = −5. Substituting:

这里 a = 2，b = 3，c = −5。代入得：

x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2)

x = (−3 ± √(9 + 40)) / 4

x = (−3 ± √49) / 4 = (−3 ± 7) / 4

x = 1 或 x = −5/2

• Write the brackets carefully: b² is 3² = 9, and −4ac means −4 × 2 × (−5) = +40.
• 注意括号：b² 是 3² = 9，而 −4ac 表示 −4 × 2 × (−5) = +40。
• The formula works for every quadratic, including those that cannot be factorised. It is a reliable fallback in the exam.
• 求根公式适用于所有二次方程，包括那些无法因式分解的式子。它是考场上的可靠后备方法。

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5. The Discriminant and the Nature of Roots | 判别式与根的性质

The expression inside the square root, b² − 4ac, is called the discriminant, often written as Δ. It tells us how many real roots the equation has without solving it:

根号内的表达式 b² − 4ac 称为判别式，通常记作 Δ。它能在不解方程的情况下告诉我们方程有几个实数根：

Δ = b² − 4ac

Discriminant | 判别式	Nature of roots | 根的性质
Δ > 0	Two distinct real roots | 两个不相等的实数根
Δ = 0	One repeated real root | 两个相等的实数根（一个重根）
Δ < 0	No real roots | 没有实数根

If Δ > 0 and is a perfect square, the equation can be solved by factorisation; if not, the roots are irrational. The discriminant also links to the graph of y = ax² + bx + c: two x-intercepts, one x-intercept, or no x-intercept.

如果 Δ > 0 且为完全平方数，则方程可用因式分解求解；否则根为无理数。判别式还与二次函数图像 y = ax² + bx + c 有关：有两个 x 轴交点、一个交点，或没有交点。

Worked example | 例题： Find the value of k for which x² + kx + 16 = 0 has exactly one real root.

For one repeated root, Δ = 0:

要有一个重根，需要 Δ = 0：

k² − 4 × 1 × 16 = 0 → k² = 64 → k = ±8

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6. Quadratic Equations in Word Problems | 二次方程应用题

Exam questions often disguise a quadratic equation as a geometry or number problem. The strategy is to define a variable, form the equation, solve it, and then check which answers are sensible in the context.

考试题常常把二次方程隐藏在几何或数字问题中。解题策略是：设变量、列方程、解方程，然后检查哪些答案在情境中合理。

Example | 示例： A rectangle has length (x + 4) cm and width (x − 2) cm. Its area is 40 cm². Find x.

示例：一个长方形的长为 (x + 4) cm，宽为 (x − 2) cm，面积为 40 cm²。求 x。

(x + 4)(x − 2) = 40

x² + 2x − 8 = 40 → x² + 2x − 48 = 0

(x + 8)(x − 6) = 0 → x = −8 或 x = 6

Since a length cannot be negative, x = −8 is rejected; the width x − 2 = 4 cm is positive, so x = 6 is the valid answer. Always reject negative solutions when they describe lengths, times or counts.

由于长度不能为负，x = −8 被舍去；此时宽 x − 2 = 4 cm 为正，所以 x = 6 是有效答案。当变量表示长度、时间或数量时，一定要舍去负数解。

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7. Common Mistakes and Exam Tips | 常见错误与考试要点

Many marks are lost in IGCSE algebra papers through avoidable errors. Here are the most common ones to watch out for:

在IGCSE数学考试中，很多分数都因可以避免的错误而丢失。以下是最常见的几个问题：

• Forgetting to set the equation to zero before factorising. The zero-product property only works when one side is 0.
• 分解因式前忘记让方程一边等于0。零积性质只在一边为0时才成立。
• Missing the ± sign when taking square roots. x² = 9 gives x = ±3, not only x = 3.
• 开平方时漏掉±号。x² = 9 的解是 x = ±3，而不只是 x = 3。
• Misreading negative coefficients in the quadratic formula, especially −4ac when c is negative.
• 在求根公式中看错负系数，尤其是当 c 为负数时 −4ac 的符号。
• Not simplifying surds in the final answer, such as √12 = 2√3. Many mark schemes require this.
• 最终答案没有化简根式，例如 √12 应化为 2√3。许多评分标准要求这样写。

In the exam, always check your solutions by substituting them back into the original equation. This takes only seconds and can rescue several marks. If factorisation does not work immediately, switch to the quadratic formula — it always gives the answer.

考试时，一定要把解代回原方程进行检验。这只需要几秒钟，却能挽回不少分数。如果因式分解一时看不出来，就改用求根公式——它总能得到答案。

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8. Practice Questions | 练习

Try these questions on your own before checking the answers. All of them are typical of IGCSE papers.

请先独立完成下面这些题目，再对照答案。它们都是IGCSE试卷中的典型题。

Q1 | 题1： Solve by factorisation: x² − 5x − 14 = 0.

Answer | 答案：x = 7 或 x = −2

Q2 | 题2： Solve by completing the square: x² − 4x − 1 = 0. Give exact answers.

Answer | 答案：x = 2 ± √5

Q3 | 题3： Use the quadratic formula to solve 3x² + x − 2 = 0.

Answer | 答案：x = 2/3 或 x = −1

Q4 | 题4： Determine the number of real roots of 2x² + 4x + 5 = 0.

Answer | 答案：Δ = 16 − 40 = −24 < 0, no real roots | 无实数根

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9. Summary | 总结

Quadratic equations are a core skill in IGCSE Mathematics. You should be able to recognise the standard form ax² + bx + c = 0, choose the most efficient method, and communicate your working clearly. Practise all three methods until they become automatic, and always use the discriminant to predict the nature of the roots before you solve.

二次方程是IGCSE数学中的核心技能。你应该能够识别标准形式 ax² + bx + c = 0，选择最有效的方法，并清晰地写出解题过程。反复练习三种方法，直到熟练掌握；在解题前，先用判别式判断根的性质。

The key idea to remember is the zero-product property for factorisation, the identity x² + bx = (x + b/2)² − (b/2)² for completing the square, and the formula x = (−b ± √(b² − 4ac)) / 2a for any quadratic. Master these, and quadratic equations will become one of your strongest topics in the exam.

需要牢记的关键是：因式分解依赖零积性质；配方依赖恒等式 x² + bx = (x + b/2)² − (b/2)²；任意二次方程可用公式 x = (−b ± √(b² − 4ac)) / 2a。掌握这些之后，二次方程将成为你考试中最有把握的内容之一。

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