Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most heavily tested topics in IGCSE Mathematics, appearing in every extended paper. They connect algebra, graphs, geometry and problem-solving. Without a solid grasp of factorisation, the quadratic formula and the discriminant, you will lose easy marks across multiple exam questions.

二次方程是 IGCSE 数学中考察频率最高的内容之一，在每一份扩展卷中都会出现。它将代数、图像、几何与应用题紧密联系在一起。如果对因式分解、求根公式和判别式掌握不牢，你会在多道考题上轻易失分。

1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is an equation of degree 2 that can be written in the form:

二次方程是次数为 2 的方程，可写成如下标准形式：

ax² + bx + c = 0, where a ≠ 0

ax² + bx + c = 0，其中 a ≠ 0

The letters a, b and c are constants (numbers), with a being the coefficient of x², b the coefficient of x, and c the constant term. The condition a ≠ 0 is essential: if a = 0, the equation becomes linear, not quadratic.

字母 a、b、c 为常数，其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。条件 a ≠ 0 至关重要：若 a = 0，方程变为一次方程而非二次方程。

Examples: x² + 5x + 6 = 0, 2x² – 3x – 2 = 0, x² – 16 = 0

例如：x² + 5x + 6 = 0，2x² – 3x – 2 = 0，x² – 16 = 0
Non-examples: x³ + 2x = 0 (cubic), 3x + 1 = 0 (linear)

反例：x³ + 2x = 0（三次方程），3x + 1 = 0（一次方程）

Before solving, always rearrange the equation so one side equals zero. This is the most common starting error students make.

在解题前，务必先整理方程，使等号一侧为零。这是学生最常犯的起始错误。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic factorises neatly into two brackets. The underlying principle is the zero product property: if ab = 0, then a = 0 or b = 0.

当二次式能够简洁地分解为两个括号相乘时，因式分解是最快捷的方法。其核心原理是零乘积性质：若 ab = 0，则 a = 0 或 b = 0。

If (x – p)(x – q) = 0, then x = p or x = q

若 (x – p)(x – q) = 0，则 x = p 或 x = q

Worked example 1: Solve x² + 5x + 6 = 0.

例题 1：解 x² + 5x + 6 = 0。

Find two numbers that multiply to 6 and add to 5: these are 2 and 3.

寻找两个数，其乘积为 6 且和为 5：这两个数是 2 和 3。

x² + 5x + 6 = (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，故 x = -2 或 x = -3。

Worked example 2: Solve 2x² – 5x – 3 = 0.

例题 2：解 2x² – 5x – 3 = 0。

Since the leading coefficient is 2, use factors of (2x + 1)(x – 3). Expanding gives 2x² – 6x + x – 3 = 2x² – 5x – 3, which is correct.

由于首项系数为 2，采用 (2x + 1)(x – 3) 的分解形式。展开得 2x² – 6x + x – 3 = 2x² – 5x – 3，与题目吻合。

2x + 1 = 0 or x – 3 = 0, so x = -½ or x = 3

Always expand your factorisation to verify it, and remember to solve after factorising, not simply leave brackets.

完成分解后务必展开验证，并记住分解后要继续求解，而不能只留下括号。

3. The Quadratic Formula | 求根公式（二次公式）

When factorisation is difficult or impossible, the quadratic formula works for every quadratic equation. You are expected to memorise it.

当因式分解困难或无法分解时，求根公式适用于一切二次方程。你需要牢记该公式。

x = (-b ± √(b² – 4ac)) / (2a)

Worked example: Solve 3x² + 7x – 5 = 0, giving your answer correct to 3 significant figures.

例题：解 3x² + 7x – 5 = 0，答案保留三位有效数字。

Here a = 3, b = 7, c = -5. Substitute into the formula:

在此 a = 3，b = 7，c = -5。代入公式：

x = (-7 ± √(7² – 4 × 3 × (-5))) / (2 × 3)

x = (-7 ± √(49 + 60)) / 6 = (-7 ± √109) / 6

Therefore x = (-7 + √109) / 6 ≈ 0.573 or x = (-7 – √109) / 6 ≈ -2.91.

因此 x = (-7 + √109) / 6 ≈ 0.573，或 x = (-7 – √109) / 6 ≈ -2.91。

The ± symbol gives two solutions: one with addition and one with subtraction. When the question asks for exact values, keep your answer in surd form with √; when it asks for decimal values, only round at the very end.

± 符号给出两个解：一个取加号，一个取减号。当题目要求精确值时，保留带根号的根式形式；当要求小数时，只在最后一步进行四舍五入。

4. 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 immediately and is also used to solve equations.

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，也常用于解方程。

x² + bx + c = (x + b/2)² – (b/2)² + c

Worked example: Solve x² + 6x + 4 = 0 by completing the square.

例题：用配方法解 x² + 6x + 4 = 0。

Take half of 6, which is 3, so (x + 3)² – 9 + 4 = 0, giving (x + 3)² – 5 = 0.

取 6 的一半，即 3，于是 (x + 3)² – 9 + 4 = 0，整理得 (x + 3)² – 5 = 0。

(x + 3)² = 5

x + 3 = ±√5, so x = -3 ± √5

For quadratics where the coefficient of x² is not 1, first factor out a from the first two terms. For example, 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2((x + 2)² – 4) + 3 = 2(x + 2)² – 5.

当 x² 的系数不为 1 时，先从首两项中提取 a。例如，2x² + 8x + 3 = 2(x² + 4x) + 3 = 2((x + 2)² – 4) + 3 = 2(x + 2)² – 5。

In the form a(x + p)² + q, the vertex is at (-p, q), and the axis of symmetry is the vertical line x = -p.

在 a(x + p)² + q 的形式中，顶点坐标为 (-p, q)，对称轴为直线 x = -p。

5. The Discriminant | 判别式

The expression b² – 4ac inside the quadratic formula is called the discriminant, often denoted by Δ. It tells us about the number and type of roots without solving the full equation.

求根公式中的 b² – 4ac 称为判别式，通常用 Δ 表示。它无需完全解方程即可判断根的数量和类型。

Δ = b² – 4ac	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 \| 一个相等的实数根（重根）
Δ < 0	No real roots \| 无实数根

Worked example: Find the value of k for which the equation x² + kx + 9 = 0 has exactly one real root.

例题：求 k 的值，使方程 x² + kx + 9 = 0 恰有一个实数根。

For one repeated root, Δ = 0. Here b = k, a = 1, c = 9.

对于重根，Δ = 0。此时 b = k，a = 1，c = 9。

k² – 4 × 1 × 9 = 0

k² = 36, so k = 6 or k = -6

The discriminant also reveals whether the graph of y = ax² + bx + c crosses the x-axis, touches it, or never meets it. This connects algebra directly to graphing questions.

判别式还能反映 y = ax² + bx + c 的图像是与 x 轴相交、相切还是永不相交。这把代数与图像题直接联系起来。

6. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, there are useful relationships that allow us to construct equations or evaluate expressions without solving.

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，存在重要的关系式，可用来构造方程或在不求解的前提下计算相关表达式。

Sum of roots: α + β = -b/a

根之和：α + β = -b/a

Product of roots: αβ = c/a

根之积：αβ = c/a

Worked example: The roots of 2x² – 6x + 3 = 0 are α and β. Find the value of α + β and αβ.

例题：设 2x² – 6x + 3 = 0 的两根为 α 和 β，求 α + β 与 αβ 的值。

α + β = -(-6)/2 = 3, αβ = 3/2

Using these, you can also form a new quadratic equation with given roots. If the required roots are r₁ and r₂, then the equation is x² – (r₁ + r₂)x + r₁r₂ = 0.

利用这些关系，还可以构造具有指定根的新二次方程。若所需根为 r₁ 和 r₂，则方程为 x² – (r₁ + r₂)x + r₁r₂ = 0。

7. Sketching Quadratic Graphs | 二次函数图像

A quadratic graph y = ax² + bx + c is a parabola. The sign of a determines its orientation: if a > 0, the parabola opens upward (a U shape); if a < 0, it opens downward (an inverted U shape).

二次函数 y = ax² + bx + c 的图像是抛物线。a 的符号决定开口方向：a > 0 时开口向上（U 形）；a < 0 时开口向下（倒 U 形）。

For a good sketch, identify three key features:

画草图时需要确定三个关键特征：

Roots: the solutions of ax² + bx + c = 0, where the curve crosses the x-axis

根：ax² + bx + c = 0 的解，即曲线与 x 轴的交点。
y-intercept: the value c, where the curve crosses the y-axis

y 轴截距：常数项 c，即曲线与 y 轴的交点。
Vertex: located at x = -b/(2a); the y-coordinate is found by substitution

顶点：x = -b/(2a)；y 坐标通过代入求得。

Worked example: Sketch y = x² – 2x – 3 by finding its roots, y-intercept and vertex.

例题：通过求根、y 轴截距和顶点，画出 y = x² – 2x – 3 的草图。

Factorise: x² – 2x – 3 = (x – 3)(x + 1), so the roots are x = 3 and x = -1. The y-intercept is c = -3. The vertex x-coordinate is -(-2)/(2 × 1) = 1, and substituting x = 1 gives y = 1 – 2 – 3 = -4, so the vertex is (1, -4).

因式分解：x² – 2x – 3 = (x – 3)(x + 1)，故根为 x = 3 和 x = -1。y 轴截距为 c = -3。顶点横坐标为 -(-2)/(2 × 1) = 1，代入 x = 1 得 y = 1 – 2 – 3 = -4，因此顶点为 (1, -4)。

The curve passes through (-1, 0), (3, 0), (0, -3) and has vertex (1, -4)

曲线经过 (-1, 0)、(3, 0)、(0, -3)，顶点为 (1, -4)

Make sure your sketch shows the correct intercepts and the axis of symmetry. In exams, marks are awarded for each labelled feature, not for artistic precision.

务必在草图中标出正确的交点与对称轴。在考试中，每个标注正确的特征都能得分，而不要求绘图的艺术精度。

8. Applications and Word Problems | 应用与文字题

Quadratic equations frequently appear in real-world contexts, especially in mensuration and kinematics. Translating words into a quadratic equation and then solving is a key IGCSE skill.

二次方程经常出现在实际情境中，尤其是测量学与运动学问题。将语言转化为二次方程再求解，是 IGCSE 的关键技能。

Worked example: A rectangle has length (x + 5) cm and width (x – 2) cm. Its area is 36 cm². Find the value of x and hence the perimeter.

例题：一个矩形的长为 (x + 5) cm，宽为 (x – 2) cm，面积为 36 cm²。求 x 的值，并进而求周长。

Area gives the equation (x + 5)(x – 2) = 36.

面积可得方程 (x + 5)(x – 2) = 36。

x² + 3x – 10 = 36

x² + 3x – 46 = 0

This does not factorise nicely, so use the formula: x = (-3 ± √(9 + 184)) / 2 = (-3 ± √193) / 2. Since length must be positive, take x = (-3 + √193) / 2 ≈ 5.45. Then length ≈ 10.45 cm, width ≈ 3.45 cm, and perimeter ≈ 2(10.45 + 3.45) ≈ 27.8 cm.

该式不易因式分解，故使用求根公式：x = (-3 ± √(9 + 184)) / 2 = (-3 ± √193) / 2。由于长度必须为正，取 x = (-3 + √193) / 2 ≈ 5.45。于是长 ≈ 10.45 cm，宽 ≈ 3.45 cm，周长 ≈ 2(10.45 + 3.45) ≈ 27.8 cm。

Always reject negative or impossible roots in word problems and state your final answer in context with the correct units.

在应用题中一定要舍去负数或不合实际的根，并带着正确的单位在情境中写出最终答案。

9. Common Mistakes and Exam Tips | 常见错误与备考提示

Many students lose marks on quadratic questions not because of difficulty, but because of small careless errors. Be aware of these pitfalls.

许多学生在二次方程题目中失分，并非因为题目难，而是由于粗心大意。请注意以下陷阱。

Forgetting to rearrange the equation to equal zero before factorising or applying the formula.

在因式分解或使用公式前，忘记将方程整理为等于零的形式。
Dividing both sides by x when x could be zero, losing the root x = 0. Instead, factorise x out.

当 x 可能为零时，直接在等式两边除以 x，从而丢失根 x = 0。应对 x 提取公因式。
Sign errors when substituting negative values of b or c into the formula.

将负数 b 或 c 代入公式时出现符号错误。
Leaving answers in unsimplified surd form when the question requires a simplified exact form.

当题目要求化简的精确值时，答案未化简为最简根式形式。
Rounding too early when the quadratic formula is involved, causing inaccurate final answers.

使用求根公式时过早四舍五入，导致最终答案不准确。
Sketching graphs without labelling roots, intercepts or the vertex.

画函数草图时，没有标注根、截距或顶点。

To revise effectively, practise every method on at least ten equations. Time yourself, and always check your solutions by substituting them back into the original equation. In the exam, show every substitution step clearly: examiners reward method marks even if your final arithmetic slips.

有效复习的方法是：每种方法至少练习十道方程。限时训练，并用回代原方程的方法验证答案。考试中，务必清晰写出每一步代入过程：即使最终计算有误，考官也会给方法分。

Quadratic equations are not just a single topic; they are a connecting thread running through the whole IGCSE syllabus. Master factorisation, the formula and the discriminant, and you will be ready for every question the exam can throw at you.

二次方程不仅仅是一个独立专题，它贯穿整个 IGCSE 数学教学大纲。掌握因式分解、求根公式和判别式，你就能从容应对考卷中的每一道相关题目。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations | 解二次方程