Quadratic Equations: Factorisation, Formula and Graphs | 二次方程：因式分解、求根公式与图像

📚 Quadratic Equations: Factorisation, Formula and Graphs | 二次方程：因式分解、求根公式与图像

Quadratic equations appear in almost every Edexcel IGCSE Mathematics paper. Whether you are expanding brackets, solving for unknown roots or sketching a parabola, a secure understanding of quadratics is essential for the Higher tier. This revision guide walks through every core skill you need, with worked examples and common exam traps.

二次方程几乎出现在每一份 Edexcel IGCSE 数学试卷中。无论你是展开括号、求解未知根，还是绘制抛物线图像，扎实掌握二次方程都是 Higher 级别考试的基础。本复习指南将带你梳理每一项核心技能，配有例题与常见考试陷阱。

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1. What Is a Quadratic Expression? | 什么是二次表达式？

A quadratic expression is a polynomial of degree 2, meaning the highest power of the variable is 2. Its general form is ax² + bx + c, where a, b and c are constants and a must not be zero. If a = 0, the expression becomes linear, not quadratic.

二次表达式是次数为 2 的多项式，即变量的最高次数为 2。它的一般形式是 ax² + bx + c，其中 a、b、c 为常数，且 a 不能为 0。若 a = 0，表达式就变成一次式而非二次式。

You will meet three distinct skills in this topic: expanding, factorising and solving. They are related, but each has its own rules. The table below summarises the vocabulary you need.

在本专题中你会遇到三种不同的技能：展开、因式分解与求解。它们相互关联，但各有各的规则。下表总结了你需要掌握的术语。

Term | 术语	Meaning | 含义
Expanding | 展开	Removing brackets by multiplying out | 通过乘法去掉括号
Factorising | 因式分解	Writing as a product of two or more brackets | 写成两个或多个括号相乘的形式
Roots | 根	Solutions of the equation f(x) = 0 | 方程 f(x) = 0 的解
Vertex | 顶点	Maximum or minimum point of the parabola | 抛物线的最高点或最低点

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2. Expanding Double Brackets | 展开双括号

To expand (x + p)(x + q), multiply each term in the first bracket by each term in the second bracket. A common method is FOIL: First, Outer, Inner, Last. For example:

要展开 (x + p)(x + q)，需要将第一个括号中的每一项分别乘以第二个括号中的每一项。常用方法是 FOIL：First（首项）、Outer（外项）、Inner（内项）、Last（末项）。例如：

(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15

The two middle terms 5x and 3x are collected together. Always check whether you can simplify the x-term before writing the final answer.

两个中间项 5x 与 3x 需要合并。在写出最终答案前，务必检查 x 项是否可以化简。

When a minus sign is involved, be extra careful with signs. Expand (x − 4)(x + 2). Using FOIL, the Outer term is +2x and the Inner term is −4x, so the coefficient of x is 2 − 4 = −2.

当涉及负号时，要格外注意符号。展开 (x − 4)(x + 2)：用 FOIL 法，外项为 +2x，内项为 −4x，所以 x 的系数为 2 − 4 = −2。

(x − 4)(x + 2) = x² − 2x − 8

If one bracket has a single term outside, for example 3(x + 2)(x − 1), expand the two brackets first, then multiply everything by 3.

如果括号外有单项式，例如 3(x + 2)(x − 1)，应先展开两个括号，再整体乘以 3。

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3. Factorising: Take out the Highest Common Factor First | 因式分解：先提最大公因式

Before attempting any quadratic factorisation, always check for a common factor in all three terms. Factorising is the reverse process of expanding: you look for what multiplies together to give the original expression.

在尝试任何二次因式分解之前，永远先检查三项是否有公因式。因式分解是展开的逆过程：你要找出相乘能得到原表达式的因式。

For example, 6x² + 9x has a common factor of 3x in both terms.

例如，6x² + 9x 的两项都有公因式 3x。

6x² + 9x = 3x(2x + 3)

You can always check your answer by expanding the bracket again. If you do not get back to the original expression, you have made a mistake.

你总可以通过重新展开括号来检验答案。如果得不到原表达式，说明你出错了。

A common error is to forget the constant term. In 4x² − 8x + 12, the HCF is 4, giving 4(x² − 2x + 3). The bracket must still contain three terms because the expression had three terms.

一个常见错误是漏掉常数项。在 4x² − 8x + 12 中，最大公因式为 4，得到 4(x² − 2x + 3)。括号内仍应有三项，因为原表达式有三项。

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4. Factorising Quadratics when a = 1 | 当 a = 1 时的二次因式分解

For x² + bx + c, you need two numbers that multiply to give c and add to give b. This is the single most important factorisation skill on the Edexcel IGCSE paper.

对于 x² + bx + c，你需要找到两个数，使之相乘等于 c，相加等于 b。这是 Edexcel IGCSE 试卷中最重要的因式分解技能。

Take x² + 5x + 6. We need two numbers whose product is 6 and sum is 5: those numbers are 2 and 3.

以 x² + 5x + 6 为例。我们需要两个数的乘积为 6，和为 5：这两个数是 2 和 3。

x² + 5x + 6 = (x + 2)(x + 3)

When c is negative, the two numbers have opposite signs. For x² − x − 12, we need a product of −12 and a sum of −1. The pair is −4 and +3.

当 c 为负数时，两个数的符号相反。对于 x² − x − 12，我们需要乘积为 −12 且和为 −1 的两个数，答案是 −4 和 +3。

x² − x − 12 = (x − 4)(x + 3)

Make a habit of listing factor pairs quickly in your working. In the exam, space is limited, so write only pairs that matter and test their sums.

养成快速列出因数对的习惯。考试时空间有限，因此只需写出相关的因数对并检验其和即可。

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5. Factorising Quadratics when a ≠ 1 | 当 a ≠ 1 时的二次因式分解

When the coefficient of x² is not 1, use the “multiply to ac, add to b” method. For 2x² + 7x + 3, here a = 2, b = 7 and c = 3, so ac = 6. We need two numbers that multiply to 6 and add to 7: they are 1 and 6.

当 x² 的系数不为 1 时，使用”相乘得 ac，相加得 b”的方法。对于 2x² + 7x + 3，这里 a = 2、b = 7、c = 3，所以 ac = 6。我们需要两个数相乘得 6、相加得 7，它们就是 1 和 6。

Split the middle term using these two numbers, then factorise in pairs.

用这两个数拆分中间项，然后进行分组因式分解。

2x² + 7x + 3 = 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

Notice that both groups, x(2x + 1) and 3(2x + 1), share the same bracket (2x + 1), which is then factored out. If the brackets do not match, you have made an error.

注意两组 x(2x + 1) 与 3(2x + 1) 含有相同的括号 (2x + 1)，随后将其提出。如果两个括号不一致，说明你出错了。

Watch out for negative coefficients. For 3x² − 5x − 2, we have ac = −6 and b = −5. The two numbers are −6 and +1.

注意负系数的情况。对于 3x² − 5x − 2，ac = −6，b = −5。两个数是 −6 和 +1。

3x² − 5x − 2 = 3x² − 6x + x − 2 = 3x(x − 2) + 1(x − 2) = (3x + 1)(x − 2)

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6. The Difference of Two Squares | 平方差公式

A special factorisation pattern occurs when the expression has exactly two terms, both perfect squares, separated by a minus sign. The rule is x² − a² = (x + a)(x − a).

当表达式恰好有两项、均为完全平方数且中间用减号连接时，会出现一种特殊的因式分解模式。公式为 x² − a² = (x + a)(x − a)。

For example, x² − 9 can be written as x² − 3², so it factorises as (x + 3)(x − 3).

例如，x² − 9 可写成 x² − 3²，因此可分解为 (x + 3)(x − 3)。

x² − 9 = (x + 3)(x − 3)

The same idea works with any constant term and with coefficients. For 4x² − 25, note that 4x² = (2x)² and 25 = 5², so 4x² − 25 = (2x + 5)(2x − 5).

同样的思路适用于任何常数项和系数。对于 4x² − 25，注意 4x² = (2x)²，25 = 5²，因此 4x² − 25 = (2x + 5)(2x − 5)。

Be careful: the sum of two squares, x² + a², cannot be factorised over the real numbers. Do not try to split x² + 9 into (x + 3)(x + 3); that expansion gives x² + 6x + 9.

要小心：两平方数之和 x² + a² 在实数范围内无法分解。不要试图把 x² + 9 拆成 (x + 3)(x + 3)；因为它的展开结果是 x² + 6x + 9。

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7. Solving Quadratic Equations by Factorisation | 用因式分解求解二次方程

A quadratic equation is a quadratic expression set equal to zero. To solve it by factorisation, first rearrange so the expression equals zero, then factorise and apply the zero product property: if A × B = 0, then A = 0 or B = 0.

二次方程就是让二次表达式等于 0 的方程。要用因式分解求解，首先要整理成表达式等于 0 的形式，然后因式分解，并应用零因子性质：若 A × B = 0，则 A = 0 或 B = 0。

Solve x² = 7x − 10. First, move every term to one side.

求解 x² = 7x − 10。首先把所有项移到同一边。

x² − 7x + 10 = 0

(x − 2)(x − 5) = 0

x = 2 or x = 5

It is essential to write “x = 2 or x = 5” as two separate solutions. A quadratic equation always has at most two real roots, and in this case both are distinct.

务必把答案写成两个独立的解 “x = 2 或 x = 5″。二次方程至多有两个实数根，本例中两个根互不相同。

If a quadratic factorises with a repeated bracket, such as x² − 6x + 9 = (x − 3)² = 0, then x = 3 is the only solution. Examiners call this a repeated root.

若二次式分解后出现重复括号，例如 x² − 6x + 9 = (x − 3)² = 0，则 x = 3 是唯一解。考官称之为重根。

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8. The Quadratic Formula | 求根公式

When a quadratic cannot be factorised easily, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by

当二次式不易因式分解时，可使用求根公式。对于 ax² + bx + c = 0，解由下式给出

x = (−b ± √(b² − 4ac)) / 2a

This formula is given on the Edexcel IGCSE formula sheet, but you must know how to substitute accurately. Solve 2x² − 4x − 3 = 0, where a = 2, b = −4 and c = −3.

这个公式会出现在 Edexcel IGCSE 公式表中，但你必须能准确代入。求解 2x² − 4x − 3 = 0，其中 a = 2、b = −4、c = −3。

x = (4 ± √(16 − 4 × 2 × (−3))) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2

Using a calculator, √10 ≈ 3.162, so x ≈ 2.581 or x ≈ −0.581. When the question asks for answers to two decimal places, write both values with the same precision.

用计算器可得 √10 ≈ 3.162，因此 x ≈ 2.581 或 x ≈ −0.581。当题目要求精确到两位小数时，两个值都要保留相同精度。

Always write the substitution step. Even if your final decimal is slightly off, the examiner can award method marks for correct substitution.

一定要写出代入步骤。即使最终小数略有偏差，考官也能因正确的代入而给方法分。

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9. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant. It tells you how many real roots a quadratic equation has without solving it. This is a frequent short-answer question on IGCSE papers.

平方根内的表达式 b² − 4ac 称为判别式。它可以在不求根的情况下判断二次方程有多少个实数根。这是 IGCSE 试卷中常见的短答题。

Discriminant Δ | 判别式 Δ	Type of Roots | 根的类型
Δ > 0	Two distinct real roots | 两个不同的实数根
Δ = 0	One repeated real root | 一个重根
Δ < 0	No real roots | 没有实数根

For example, the equation x² + 4x + 5 = 0 has Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4, which is negative. Therefore it has no real roots, so its graph does not cross the x-axis.

例如，方程 x² + 4x + 5 = 0 的判别式 Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4，为负数。因此它没有实数根，其图像不与 x 轴相交。

Some questions ask you to find the value of k that gives a repeated root. Set b² − 4ac = 0 and solve for the unknown constant.

有些题目要求你求出使方程有重根的 k 值。令 b² − 4ac = 0，然后解出未知常数即可。

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10. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This form reveals the vertex of the parabola and is sometimes the fastest way to solve an equation.

配方法将二次式改写为 (x + p)² + q 的形式。这种形式能直接显示抛物线的顶点，有时也是求解方程的最快途径。

Take x² + 6x + 2. Half of 6 is 3, so we write (x + 3)². Expanding (x + 3)² gives x² + 6x + 9, which is 7 more than our expression, so we subtract 7.

以 x² + 6x + 2 为例。6 的一半是 3，因此先写 (x + 3)²。展开 (x + 3)² 得到 x² + 6x + 9，比原表达式多了 7，所以要减去 7。

x² + 6x + 2 = (x + 3)² − 7

The completed square form (x + 3)² − 7 tells us the minimum point is at (−3, −7). To solve x² + 6x + 2 = 0, rearrange and square root both sides.

配方形式 (x + 3)² − 7 告诉我们最低点为 (−3, −7)。要解 x² + 6x + 2 = 0，可移项后两边开平方。

(x + 3)² = 7 ⇒ x + 3 = ±√7 ⇒ x = −3 ± √7

Remember the plus-or-minus sign. Taking only the positive square root would lose one solution.

切记要写正负号。如果只取正的平方根，就会丢失一个解。

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11. Sketching Quadratic Graphs | 二次函数图像的绘制

A quadratic graph y = ax² + bx + c is a parabola. If a > 0, it is U-shaped with a minimum point. If a < 0, it is n-shaped with a maximum point.

二次函数图像 y = ax² + bx + c 是抛物线。若 a > 0，则为 U 形，有最低点；若 a < 0，则为 ∩ 形，有最高点。

To sketch y = x² − 4x + 3, find three key features. First, factorise to find the roots.

要绘制 y = x² − 4x + 3 的草图，需要找到三个关键特征。首先，因式分解求根。

y = (x − 1)(x − 3), so roots are x = 1 and x = 3

Second, find the y-intercept by substituting x = 0, giving y = 3. Third, the vertex lies halfway between the roots at x = 2, and substituting gives y = −1.

其次，代入 x = 0 求 y 截距，得到 y = 3。第三，顶点位于两根中间 x = 2 处，代入得 y = −1。

The completed square form x² − 4x + 3 = (x − 2)² − 1 confirms the vertex at (2, −1). Label the roots, the y-intercept and the vertex clearly on your sketch.

配方形式 x² − 4x + 3 = (x − 2)² − 1 确认顶点为 (2, −1)。在草图上清楚标出根、y 截距和顶点。

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12. Exam Tips and Common Mistakes | 考试技巧与常见错误

Below are the traps that lose marks most often on Edexcel IGCSE quadratic questions, together with tips to avoid them.

以下是在 Edexcel IGCSE 二次方程问题中最常见的失分陷阱，以及规避方法。

• Never divide both sides by x. If x² = 5x, dividing by x gives x = 5 and loses the solution x = 0. Instead rearrange: x² − 5x = 0, then x(x − 5) = 0.

  决不能两边同除以 x。若 x² = 5x，除以 x 只会得到 x = 5，从而丢失 x = 0 这个解。正确做法是移项：x² − 5x = 0，然后 x(x − 5) = 0。

• Always rearrange the equation to equal zero before factorising. Solving x² − 4x = 12 by splitting it as (x − 6)(x + 2) = 12 is invalid because the zero product property only works when the product equals zero.

  在因式分解前，一定要先把方程整理成等于 0 的形式。把 x² − 4x = 12 拆成 (x − 6)(x + 2) = 12 是错误的，因为零因子性质只有在乘积等于 0 时才能使用。

• Check signs carefully when substituting into the quadratic formula. The expression −b is positive when b is negative, and the term −4ac becomes positive when both a and c are negative.

  代入求根公式时要仔细检查符号。b 为负数时 −b 为正；a、c 均为负数时，−4ac 会变为正数。

• Read the required accuracy. If the question says “give your answers correct to 2 decimal places”, do not leave surds such as √10 in your final answer.

  看清精度要求。如果题目要求”答案精确到两位小数”，不要在最终答案中保留 √10 这类根式。

• When sketching, show the line of symmetry. The vertex x-coordinate is always the midpoint of the two roots, which is also the turning point of the parabola.

  绘制草图时，要画出对称轴。顶点的 x 坐标始终是两根的中点，也是抛物线的转折点。

Practise every method until it becomes automatic. Factorisation is quick, the formula is reliable for everything, completing the square connects nicely to graph sketches, and the discriminant helps you check whether your answers make sense.

请反复练习每一种方法，直到运用自如。因式分解速度快，求根公式对任何情况都可靠，配方法与图像绘制紧密相连，而判别式则能帮助你检验答案是否合理。

With regular practice, quadratic questions will become some of the easiest marks on your paper. Try ten mixed questions without looking at the answers, then mark them honestly and revisit any topic where you lost marks.

只要坚持练习，二次方程问题会成为你试卷中最容易拿分的部分之一。试着不看答案完成十道混合题，然后诚实地批改，并针对失分点重新复习。

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