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Solving Quadratic Equations | 解二次方程

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📚 Solving Quadratic Equations | 解二次方程

In IGCSE Mathematics, quadratic equations are a core topic for the Extended curriculum, appearing in both Paper 2 and Paper 4. Mastering the techniques of factorisation, completing the square, and using the quadratic formula is essential for high marks. This revision guide breaks down each method step by step, with worked examples and common pitfalls explained clearly.

在 IGCSE 数学中，二次方程是扩展（Extended）课程的核心内容，在 Paper 2 和 Paper 4 中都会出现。掌握因式分解、配方以及求根公式这三大方法，是取得高分的关键。本复习指南将逐一讲解每种方法，配有完整的解题步骤，并明确指出常见的失分陷阱。

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

A quadratic equation is a polynomial equation whose highest power of the unknown variable is 2. The general form is

二次方程是未知数最高次数为 2 的多项式方程，其一般形式为

ax² + bx + c = 0, where a ≠ 0

Here a, b, and c are real numbers, and a is called the leading coefficient. If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

其中 a、b、c 是实数，a 称为首项系数。如果 a = 0，方程就变成一次方程，因此 a ≠ 0 这一条件至关重要。

  • x² − 4x + 3 = 0 is quadratic, with a = 1, b = −4, c = 3.
  • x² − 4x + 3 = 0 是二次方程，其中 a = 1，b = −4，c = 3。
  • 2x² + 5 = 0 is also quadratic; here b = 0.
  • 2x² + 5 = 0 也是二次方程，此时 b = 0。
  • x³ + 2x − 1 = 0 is not quadratic because the highest power is 3.
  • x³ + 2x − 1 = 0 不是二次方程，因为最高次数是 3。

2. Writing Equations in Standard Form | 化为标准形式

Before solving, every quadratic equation must be rearranged into the form ax² + bx + c = 0. This means collecting all terms on one side of the equals sign and simplifying.

在求解之前，必须先把每个二次方程整理成 ax² + bx + c = 0 的形式，也就是把所有项移到等号的一边并化简。

Example: rearrange 2x² + 3 = 5x.

例：将 2x² + 3 = 5x 化为标准形式。

2x² − 5x + 3 = 0

Now a = 2, b = −5, c = 3. Always double-check the signs of the coefficients after moving terms across the equals sign.

现在 a = 2，b = −5，c = 3。移项之后务必检查各项系数的符号。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic expression factorises nicely. The key principle is the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

当二次式能够顺利分解时，因式分解法是最快捷的方法。其核心原理是零积性质：若两个表达式的乘积为零，则其中至少有一个为零。

Step-by-step method | 步骤：

  • Write the equation in the form ax² + bx + c = 0.
  • 将方程写成 ax² + bx + c = 0 的形式。
  • Factorise the left-hand side into two brackets.
  • 将左边分解成两个括号相乘的形式。
  • Set each bracket equal to zero and solve the resulting linear equations.
  • 令每个括号等于零，解出两个一次方程。

Example 1: solve x² + 5x + 6 = 0. Find two numbers that multiply to 6 and add to 5, namely 2 and 3.

例 1：解 x² + 5x + 6 = 0。找两个数，相乘得 6 且相加得 5，即 2 和 3。

(x + 2)(x + 3) = 0

Hence x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.

因此 x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。

Example 2: solve 2x² − 8x = 0. Here the common factor is 2x.

例 2：解 2x² − 8x = 0。这里公因式是 2x。

2x(x − 4) = 0

So 2x = 0 or x − 4 = 0, giving x = 0 or x = 4. Remember: when an equation has no constant term, always factor out x rather than dividing both sides by x, otherwise you lose the root x = 0.

所以 2x = 0 或 x − 4 = 0，得到 x = 0 或 x = 4。注意：当方程没有常数项时，一定要提取公因式 x，而不是两边同除以 x，否则会丢失 x = 0 这个根。

Example 3: solve x² − 9 = 0 using the difference of two squares.

例 3：利用平方差公式解 x² − 9 = 0。

(x − 3)(x + 3) = 0, so x = ±3

For any expression a² − b², the factorised form is (a − b)(a + b).

对于任意 a² − b²，其因式分解形式为 (a − b)(a + b)。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method works for any quadratic, including those that do not factorise, and it is also used to find turning points of graphs.

配方法将二次式改写为 (x + p)² + q 的形式。这种方法适用于任何二次方程，包括无法因式分解的情形，同时也常用于求抛物线图像的顶点。

Example: solve x² + 6x − 7 = 0.

例：解 x² + 6x − 7 = 0。

First, halve the coefficient of x, which is 3, and write (x + 3)². Then subtract 3² to keep the expression equivalent.

首先，把 x 的系数除以 2 得到 3，写成 (x + 3)²，然后减去 3² 以保持原式不变。

(x + 3)² − 9 − 7 = 0

Simplify to (x + 3)² = 16, then take the square root of both sides.

化简得 (x + 3)² = 16，然后两边同时开平方。

x + 3 = ±4

Therefore x = 1 or x = −7. Note the ± sign: taking a square root always gives both a positive and a negative result.

因此 x = 1 或 x = −7。注意 ± 号：开平方的结果总是正负两个值。

5. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula provides a reliable solution for any equation of the form ax² + bx + c = 0.

当因式分解困难或无法分解时，求根公式为 ax² + bx + c = 0 形式的任何方程提供了可靠的解法。

x = (−b ± √(b² − 4ac)) / (2a)

Example: solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute carefully into the formula.

例：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入公式时需格外小心。

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

The discriminant is 16 + 24 = 40, so the solutions are

判别式为 16 + 24 = 40，所以解为

x = (4 ± √40) / 4 = (4 ± 2√10) / 4 = (2 ± √10) / 2

In decimal form, x ≈ 2.581 or x ≈ −0.581. Always write the exact answer using surds first, then give a decimal approximation if required.

化为小数时，x ≈ 2.581 或 x ≈ −0.581。解题时应先用根式表示精确答案，再按要求给出近似值。

6. The Discriminant | 判别式

The expression Δ = b² − 4ac under the square root is called the discriminant. It tells us the number and nature of the roots without solving the equation fully.

根号内的表达式 Δ = b² − 4ac 称为判别式。不需要解方程，由它即可判断根的个数和性质。

Discriminant | 判别式 Nature of Roots | 根的性质 Graph Meaning | 图像含义
Δ > 0 Two distinct real roots | 两个不等实根 The parabola crosses the x-axis at two points | 抛物线与 x 轴有两个交点
Δ = 0 One repeated root | 一个重根 The parabola touches the x-axis at one point | 抛物线与 x 轴相切于一点
Δ < 0 No real roots | 无实根 The parabola does not meet the x-axis | 抛物线与 x 轴没有交点

Example: determine the number of roots of x² − 6x + 10 = 0. Here Δ = (−6)² − 4 × 1 × 10 = 36 − 40 = −4 < 0, so the equation has no real solutions.

例：判断 x² − 6x + 10 = 0 的根的个数。这里 Δ = (−6)² − 4 × 1 × 10 = 36 − 40 = −4 < 0，因此方程无实数解。

7. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the following relationships hold. These are useful for verifying answers and solving problems without factorising.

对于二次方程 ax² + bx + c = 0，若其两根为 α 和 β，则有以下关系。这些关系可用于检验答案，也可在不分解的情况下解题。

α + β = −b / a and αβ = c / a

Example: for 2x² − 8x + 6 = 0, the roots are 1 and 3. Their sum is 4, and −b/a = 8/2 = 4, which matches. Their product is 3, and c/a = 6/2 = 3, which also matches.

例：对于 2x² − 8x + 6 = 0，两根为 1 和 3。它们的和为 4，而 −b/a = 8/2 = 4，结果一致。它们的积为 3，而 c/a = 6/2 = 3，同样一致。

8. Solving Word Problems | 解应用题

Quadratic equations often appear in geometry and number problems. The strategy is to define the unknown, form the equation from the given information, solve it, and then check whether each solution makes sense in the original context.

二次方程经常出现在几何和数字问题中。解题策略是：设未知数，根据已知条件建立方程，求解，然后检验每个解在原问题情境中是否合理。

Example: a rectangle is 3 cm longer than it is wide, and its area is 40 cm². Find its width.

例：一个长方形的长比宽多 3 cm，面积为 40 cm²。求其宽。

Let the width be w cm. Then the length is w + 3, so the area gives the equation

设宽为 w cm，则长为 w + 3，由面积得方程

w(w + 3) = 40

Expanding gives w² + 3w − 40 = 0, which factorises as (w + 8)(w − 5) = 0. Hence w = −8 or w = 5. Since a width cannot be negative, we reject w = −8 and conclude that the width is 5 cm.

展开得 w² + 3w − 40 = 0，因式分解为 (w + 8)(w − 5) = 0。因此 w = −8 或 w = 5。由于宽度不能为负数，舍去 w = −8，故宽为 5 cm。

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many marks are lost in quadratic questions due to small but repeated errors. Here are the most common ones and how to avoid them.

在二次方程题目中，许多分数都是因为一些重复出现的小错误而丢失的。以下是最常见的错误及其避免方法。

  • Dividing by x: never divide both sides by x, because this eliminates the root x = 0. Factorise instead.
  • 两边同除以 x：切勿两边同除以 x，因为这样会丢失根 x = 0。应使用因式分解。
  • Forgetting ±: when taking the square root of both sides, always write ± on one side.
  • 忘记 ± 号：两边同时开平方时，必须在一侧写上 ± 号。
  • Sign errors: when substituting b into the quadratic formula, include its sign. In 2x² − 4x − 3 = 0, b is −4, not 4.
  • 符号错误：把 b 代入求根公式时，务必包含其符号。在 2x² − 4x − 3 = 0 中，b 是 −4，而不是 4。
  • Not rearranging: always rewrite the equation in the form ax² + bx + c = 0 before applying any method.
  • 未化为标准形式：使用任何方法前，务必先把方程改写为 ax² + bx + c = 0 的形式。

10. Practice Questions | 练习

Attempt these questions without looking at the answers, then check your working carefully.

先独立完成以下题目，再仔细检查你的解题过程。

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