Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics Paper 2. This revision guide covers every essential method — factorisation, the quadratic formula, completing the square and graph sketching — with exam-style examples and common pitfalls explained step by step.

二次方程是 IGCSE 数学 Paper 2 中最高频的考点之一。本复习指南全面覆盖所有核心方法——因式分解、二次公式、配方法以及图像画法——并配有考试风格例题和常见误区的分步讲解。

1. The Standard Form | 标准形式

Every quadratic equation can be written in the standard form

ax² + bx + c = 0, where a ≠ 0

Here a is the coefficient of x², b is the coefficient of x, and c is the constant term. For example, 3x² − 5x + 2 = 0 has a = 3, b = −5 and c = 2.

每个二次方程都可以写成标准形式

ax² + bx + c = 0，其中 a ≠ 0

这里 a 是 x² 的系数，b 是 x 的系数，c 是常数项。例如，3x² − 5x + 2 = 0 中 a = 3，b = −5，c = 2。

You must always rearrange an equation into this form before solving. A very common exam error is trying to factorise while the right-hand side is not zero.

在求解之前，你必须先将方程整理成这种标准形式。一个非常常见的考试错误是在等号右边不为零时就直接进行因式分解。

2. Expanding Double Brackets | 展开双重括号

Before factorising, you must be confident expanding expressions of the form (x + p)(x + q). Use the distributive law or the FOIL method: multiply First, Outer, Inner, Last terms.

在因式分解之前，你必须熟练掌握 (x + p)(x + q) 形式的展开。使用分配律或 FOIL 方法：依次相乘 First（首项）、Outer（外项）、Inner（内项）、Last（末项）。

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

Notice that 8 = 3 + 5 and 15 = 3 × 5. This pattern is the key to factorisation.

注意 8 = 3 + 5，而 15 = 3 × 5。这个规律正是因式分解的关键。

3. Factorising Quadratics of the Form x² + bx + c | 因式分解 x² + bx + c 形式的二次式

To factorise x² + bx + c, find two integers whose sum is b and whose product is c.

要对 x² + bx + c 进行因式分解，找出两个整数，使它们的和为 b，乘积为 c。

x² + 7x + 12 = (x + 3)(x + 4)

Check: 3 + 4 = 7 and 3 × 4 = 12. If the constant term is negative, one factor must be negative. For example:

验证：3 + 4 = 7，3 × 4 = 12。如果常数项为负，则两个因子中必须有一个为负。例如：

x² − x − 6 = (x − 3)(x + 2)

Here −3 + 2 = −1 and −3 × 2 = −6. Always expand your answer to check that it matches the original expression.

这里 −3 + 2 = −1，−3 × 2 = −6。始终将你的答案展开验证，确保与原式一致。

4. Solving by Factorisation | 用因式分解求解

The zero-product property states that if A × B = 0, then either A = 0 or B = 0. This provides the direct route from factorised quadratic to solution.

零乘积性质指出：如果 A × B = 0，那么 A = 0 或 B = 0。这为从因式分解后的二次式直接求解提供了途径。

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3

Both solutions are valid. In IGCSE you should always write the two solutions clearly, usually as ‘x = 2 or x = 3’. Never stop after factorising — you must then solve each bracket equal to zero.

两个解都有效。在 IGCSE 考试中，你应该清晰写出两个解，通常写成“x = 2 或 x = 3”。绝不能在因式分解后停下——你还必须令每个括号等于零并求解。

5. The Quadratic Formula | 二次公式

When factorisation is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0,

当因式分解困难或无法进行时，使用二次公式。对于 ax² + bx + c = 0，

x = (−b ± √(b² − 4ac)) / 2a

This formula is given in the IGCSE formula sheet, but you must know how to substitute values carefully. Solve 2x² + 3x − 5 = 0:

该公式在 IGCSE 公式表中给出，但你必须知道如何仔细代入数值。解 2x² + 3x − 5 = 0：

x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2) = (−3 ± √49) / 4 = (−3 ± 7) / 4

So x = (−3 + 7)/4 = 1 or x = (−3 − 7)/4 = −2.5. Use a calculator to evaluate the square root, and give exact answers where requested.

因此 x = (−3 + 7)/4 = 1 或 x = (−3 − 7)/4 = −2.5。使用计算器求平方根，并在题目要求时给出精确答案。

6. Completing the Square | 配方法

Completing the square rewrites x² + bx + c in the form (x + p)² + q. The key fact is that p = b/2.

配方法将 x² + bx + c 改写成 (x + p)² + q 的形式。关键是 p = b/2。

x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7

To solve, set the completed-square expression to zero:

要求解，令配方法后的表达式为零：

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

This method also reveals the turning point of the graph. For y = (x + 3)² − 7, the minimum point is at (−3, −7).

这种方法还能揭示图像的顶点。对于 y = (x + 3)² − 7，最低点为 (−3, −7)。

7. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant, often written Δ. It determines the nature of the roots.

二次公式中的表达式 b² − 4ac 称为判别式，通常记作 Δ。它决定了根的性质。

Discriminant Δ	Nature of roots	根的性质
Δ > 0	two distinct real roots	两个不同的实数根
Δ = 0	one repeated real root	一个重根（两相等实根）
Δ < 0	no real roots	无实数根

For example, x² + x + 1 = 0 has Δ = 1 − 4 = −3 < 0, so it has no real roots. The graph never crosses the x-axis.

例如，x² + x + 1 = 0 的 Δ = 1 − 4 = −3 < 0，因此没有实数根。图像永远不会与 x 轴相交。

8. Quadratic Graphs | 二次函数图像

The graph of y = ax² + bx + c is a parabola. When a > 0 it opens upward with a minimum point; when a < 0 it opens downward with a maximum point.

y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时开口向上，有最低点；当 a < 0 时开口向下，有最高点。

Useful features to identify when sketching:

画图时需要确定的关键特征：

Roots: solve ax² + bx + c = 0 — these are the x-intercepts.

根：解 ax² + bx + c = 0——这些是 x 轴交点。
y-intercept: the point (0, c) where the curve crosses the y-axis.

y 轴截距：曲线与 y 轴相交于点 (0, c)。
Turning point: found by completing the square, y = a(x − h)² + k gives vertex (h, k).

顶点：通过配方法求得，y = a(x − h)² + k 的顶点为 (h, k)。

Sketching a fully labelled graph is a common 3–4 mark question in Paper 2.

画出带有完整标注的图像是 Paper 2 中常见的 3–4 分题目。

9. Word Problems | 应用题

Quadratics often appear in geometric and motion contexts. Read the problem carefully, define a variable, form a quadratic equation, and check which solution makes sense.

二次方程常出现在几何和运动问题中。仔细读题、设变量、建立二次方程，并检查哪一个解是合理的。

Example: A rectangle has length (x + 5) cm and width (x − 2) cm. Its area is 50 cm². Find x.

例如：一个长方形的长为 (x + 5) cm，宽为 (x − 2) cm，面积为 50 cm²。求 x。

(x + 5)(x − 2) = 50 → x² + 3x − 10 = 50 → x² + 3x − 60 = 0

Using the quadratic formula gives x ≈ 6.27 or x ≈ −9.27. Since lengths cannot be negative, x = 6.27 is the only valid answer.

使用二次公式得 x ≈ 6.27 或 x ≈ −9.27。由于长度不能为负，x = 6.27 是唯一有效答案。

10. Common Exam Mistakes | 常见考试错误

Being aware of typical errors can save valuable marks. Here are the most frequent mistakes students make:

了解典型错误可以帮你挽回宝贵分数。以下是学生最常犯的错误：

Solving 2x² = 8x by dividing both sides by x, losing the solution x = 0. Correct: 2x² − 8x = 0 → 2x(x − 4) = 0.

解 2x² = 8x 时两边除以 x，从而丢失解 x = 0。正确做法：2x² − 8x = 0 → 2x(x − 4) = 0。
Forgetting to rearrange to the form ax² + bx + c = 0 before using the quadratic formula.

在使用二次公式前忘记整理成 ax² + bx + c = 0 的形式。
Sign errors when substituting negative values of b or c into the formula.

将负的 b 或 c 代入公式时出现符号错误。
Stating only one solution when two distinct roots exist.

当存在两个不同实根时只写出一个解。
Writing ± incorrectly when taking square roots in completing the square. Remember to include the negative root.

在配方法中开平方时错误使用 ±。记住要包含负根。

11. Practice Questions | 练习题

Test yourself with these exam-style questions before moving on:

在继续之前，用这些考试风格的题目测试自己：

Solve x² − 9x + 14 = 0 by factorisation.

用因式分解解 x² − 9x + 14 = 0。
Solve 3x² + 5x − 2 = 0 using the quadratic formula. Give answers to 2 decimal places.

用二次公式解 3x² + 5x − 2 = 0，答案保留两位小数。
Find the discriminant of 4x² − 4x + 1 = 0 and state the nature of its roots.

求 4x² − 4x + 1 = 0 的判别式，并说明根的性质。
Write x² − 8x + 3 in the form (x − p)² + q, and hence state the minimum point of the curve y = x² − 8x + 3.

将 x² − 8x + 3 写成 (x − p)² + q 的形式，并由此写出曲线 y = x² − 8x + 3 的最低点。

Answers: 1) x = 2 or x = 7. 2) x = 0.33 or x = −2.00. 3) Δ = 0, one repeated real root. 4) (x − 4)² − 13, minimum point (4, −13).

答案：1) x = 2 或 x = 7。2) x = 0.33 或 x = −2.00。3) Δ = 0，一个重根。4) (x − 4)² − 13，最低点 (4, −13)。

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Solving Quadratic Equations | 解二次方程