Mastering Quadratic Equations for IGCSE Mathematics | 掌握二次方程：IGCSE数学核心专题

📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握二次方程：IGCSE数学核心专题

Quadratic equations are among the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, graphs, and many examination questions. This guide explains every essential skill you need, from factorisation to the quadratic formula, with clear worked examples.

二次方程是IGCSE数学中最核心的内容之一。它出现在代数、坐标几何、函数图像以及大量考试题目中。本指南将详细讲解你所需的全部关键技巧，从因式分解到公式法，并配有清晰的例题。

1. Standard Form and Key Concepts | 标准形式与关键概念

Every quadratic equation can be written in a standard form. The standard form is the most useful starting point for solving, sketching, and analysing quadratics.

每一个二次方程都可以写成标准形式。标准形式是求解、画图和讨论二次函数性质的最佳起点。

English: A quadratic equation in one variable is any equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

中文：含一个未知数的二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。
English: If a = 0, the equation becomes linear, not quadratic. For example, 3x² + 2x – 1 = 0 is quadratic, while 2x + 5 = 0 is linear.

中文：如果 a = 0，方程就变成一次方程，而不是二次方程。例如 3x² + 2x – 1 = 0 是二次方程，而 2x + 5 = 0 是一次方程。
English: The solutions of a quadratic equation are often called roots. A quadratic equation can have two real roots, one repeated root, or no real roots.

中文：二次方程的解通常被称为根。二次方程可能有两个不同实根、一个重根，或者没有实根。

For example, the equation x² – 3x + 2 = 0 has roots x = 1 and x = 2. You should always check whether a given equation is already in standard form before deciding on a method.

例如，方程 x² – 3x + 2 = 0 的根是 x = 1 和 x = 2。在决定使用哪种解法之前，你应该先确认方程是否已经写成标准形式。

2. Solving by Factorisation | 因式分解法求解

Factorisation is often the quickest method for solving quadratic equations, especially when the coefficients are simple.

因式分解通常是求解二次方程最快的方法，尤其是当系数比较简单时。

English: The key idea is the zero product property: if p × q = 0, then either p = 0 or q = 0.

中文：核心思想是零积性质：如果 p × q = 0，那么必有 p = 0 或 q = 0。

x² – 5x + 6 = 0 ⇒ (x – 2)(x – 3) = 0 ⇒ x = 2 or x = 3

English: To factorise x² + bx + c, find two numbers that multiply to give c and add to give b. For x² – 5x + 6, the numbers are -2 and -3 because (-2) × (-3) = 6 and (-2) + (-3) = -5.

中文：要分解 x² + bx + c，需要找到两个数，它们相乘等于 c，相加等于 b。对于 x² – 5x + 6，这两个数是 -2 和 -3，因为 (-2) × (-3) = 6，且 (-2) + (-3) = -5。

English: When the coefficient of x² is not 1, for example 2x² + 7x + 3 = 0, you need to split the middle term. Multiply 2 and 3 to get 6, then find two numbers that multiply to 6 and add to 7. These are 1 and 6, so the expression becomes 2x² + x + 6x + 3.

中文：当 x² 的系数不为 1 时，例如 2x² + 7x + 3 = 0，需要使用拆项法。将 2 和 3 相乘得到 6，然后找到两个数相乘为 6、相加为 7。这两个数是 1 和 6，所以原式改写为 2x² + x + 6x + 3。

English: Group the terms: (2x² + x) + (6x + 3) = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1).

中文：分组得 (2x² + x) + (6x + 3) = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1)。
English: Therefore 2x² + 7x + 3 = 0 has roots x = -3 and x = -½.

中文：因此 2x² + 7x + 3 = 0 的根为 x = -3 和 x = -½。

Always expand your factors to check your work. A quick expansion can prevent careless errors.

务必通过展开因式来检查结果。快速展开可以避免粗心的错误。

3. Solving by Completing the Square | 配方法求解

Completing the square is a powerful technique that works for every quadratic equation. It also helps you find the turning point of a quadratic graph.

配方法是一种通用且强大的技巧，适用于所有二次方程。它还能帮助你找到二次函数图像的顶点。

English: Use the identity x² + bx = (x + b/2)² – (b/2)². Add or subtract the constant term carefully.

中文：使用恒等式 x² + bx = (x + b/2)² – (b/2)²。处理常数项时要特别小心。

x² + 6x + 2 = 0 ⇒ (x + 3)² – 9 + 2 = 0 ⇒ (x + 3)² = 7

English: Take the square root of both sides. Since squaring can be reversed by a positive or negative root, you must include the ± sign.

中文：对两边开平方。因为平方可以由正根或负根还原，所以一定要保留 ± 号。

x + 3 = ±√7 ⇒ x = -3 ± √7

English: If the coefficient of x² is not 1, divide the whole equation by that coefficient first. For example 2x² + 8x – 10 = 0 becomes x² + 4x – 5 = 0.

中文：如果 x² 的系数不是 1，先将整个方程除以该系数。例如 2x² + 8x – 10 = 0 化简为 x² + 4x – 5 = 0。
English: Then complete the square: (x + 2)² – 4 – 5 = 0, so x = -2 ± 3, giving x = 1 or x = -5.

中文：然后配方：(x + 2)² – 4 – 5 = 0，所以 x = -2 ± 3，即 x = 1 或 x = -5。

Completing the square is also useful when a quadratic has irrational roots, because the answer remains in exact form, such as x = -3 ± √7.

当二次方程的根是无理数时，配方法尤其有用，因为答案可以保持精确形式，例如 x = -3 ± √7。

4. Solving by the Quadratic Formula | 公式法求解

The quadratic formula is the most general method. You can use it for any quadratic equation, even when factorisation is difficult or impossible.

公式法是最通用的解法。你可以用它求解任何二次方程，即使因式分解困难或无法因式分解。

x = (-b ± √(b² – 4ac)) / 2a

English: To use this formula, write the equation in the form ax² + bx + c = 0, then substitute the values of a, b and c carefully.

中文：使用该公式前，先将方程写成 ax² + bx + c = 0 的形式，再仔细代入 a、b、c 的值。

Example: Solve 2x² – 3x – 2 = 0. Here a = 2, b = -3, c = -2.

例如：求解 2x² – 3x – 2 = 0。此时 a = 2，b = -3，c = -2。

x = (-(-3) ± √((-3)² – 4×2×(-2))) / (2×2) = (3 ± √25) / 4 = (3 ± 5) / 4

English: The two solutions are x = (3 + 5)/4 = 2 and x = (3 – 5)/4 = -½.

中文：两个解为 x = (3 + 5)/4 = 2 和 x = (3 – 5)/4 = -½。
English: Always check that b² – 4ac is non-negative before writing down real solutions. If it is negative, the equation has no real roots.

中文：写出实数解之前，务必检查 b² – 4ac 是否为非负数。如果它是负数，说明方程没有实数根。

The formula is worth remembering because it works every time, but you must also know when it is better not to use it. Factorisation is often faster.

这个公式值得牢记，因为它永远有效。但你也要知道什么时候不需要用它，因为因式分解往往更快。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root, usually written as Δ or simply D. It tells you how many real roots a quadratic equation has.

判别式是公式法中根号内的部分，通常记为 Δ 或 D。它可以告诉你二次方程有多少个实根。

Δ = b² – 4ac

English: If Δ > 0, there are two distinct real roots. For example, x² – 4x + 1 = 0 has Δ = 12, so it has two real roots.

中文：如果 Δ > 0，则有两个不同的实根。例如 x² – 4x + 1 = 0 的 Δ = 12，所以它有两个实根。
English: If Δ = 0, there is exactly one repeated real root. For example, x² + 2x + 1 = 0 has Δ = 0, so its only root is x = -1.

中文：如果 Δ = 0，则只有一个重根。例如 x² + 2x + 1 = 0 的 Δ = 0，所以它唯一的根是 x = -1。
English: If Δ < 0, there are no real roots. For example, x² + 2x + 3 = 0 has Δ = -8, so its graph never crosses the x-axis.

中文：如果 Δ < 0，则没有实数根。例如 x² + 2x + 3 = 0 的 Δ = -8，所以它的图像不会穿过 x 轴。

Exam questions often ask you to find the range of k for which an equation has real roots. Set the discriminant greater than or equal to zero and solve the resulting inequality.

考试中常要求你找出使方程有实根时 k 的取值范围。这时需要令判别式大于等于零，再解所得的不等式。

6. Sum and Product of Roots | 根与系数的关系

For quadratic equations, there is an elegant relationship between the coefficients and the roots. This can save you time in algebra and graph questions.

对于二次方程，系数与根之间存在简洁的关系。这类关系可以节省你在代数和图像题中的时间。

English: If α and β are the roots of ax² + bx + c = 0, then the sum of the roots is α + β = -b/a and the product is αβ = c/a.

中文：如果 α 和 β 是方程 ax² + bx + c = 0 的根，那么两根之和为 α + β = -b/a，两根之积为 αβ = c/a。

Example: For 3x² – 6x + 2 = 0, the sum of roots is -(-6)/3 = 2 and the product is 2/3.

例如：对于 3x² – 6x + 2 = 0，两根之和为 -(-6)/3 = 2，两根之积为 2/3。

English: If you know the roots α and β, you can reconstruct the quadratic equation as x² – (α + β)x + αβ = 0.

中文：如果已知两根 α 和 β，可以还原二次方程为 x² – (α + β)x + αβ = 0。

English: These relationships are useful when you need to form a new equation from given roots without solving the original equation.

中文：当需要根据已知根构造新方程，而不需要解原方程时，这些关系非常有用。
English: For example, if a quadratic has roots 3 and -4, then the equation is x² – (3 + (-4))x + 3×(-4) = 0, which is x² + x – 12 = 0.

中文：例如，如果一个二次方程的两根为 3 和 -4，则方程为 x² – (3 + (-4))x + 3×(-4) = 0，即 x² + x – 12 = 0。

You can always check your result by substituting the roots back into the equation.

你总可以通过回代根来检查结果是否正确。

7. Sketching Quadratic Graphs | 二次函数图像作图

A quadratic function y = ax² + bx + c produces a curve called a parabola. The sign of a determines whether the parabola opens upwards or downwards.

二次函数 y = ax² + bx + c 的图像是一条抛物线。a 的正负决定抛物线开口向上还是向下。

English: If a > 0, the parabola has a minimum point and opens upwards, like the graph of y = x².

中文：如果 a > 0，抛物线向上开口，有最低点，例如 y = x² 的图像。
English: If a < 0, the parabola has a maximum point and opens downwards, like the graph of y = -x².

中文：如果 a < 0，抛物线向下开口，有最高点，例如 y = -x² 的图像。

English: To sketch a quadratic graph, find the x-intercepts by solving y = 0, find the y-intercept by setting x = 0, and find the vertex by completing the square or using x = -b/(2a).

中文：要画二次函数草图，需要解 y = 0 找到 x 轴交点，令 x = 0 找到 y 轴截距，并通过配方或 x = -b/(2a) 找到顶点。

Example: Sketch y = x² – 4x + 3.

示例：画出 y = x² – 4x + 3 的草图。

English: Factorise y = (x – 1)(x – 3), so the curve crosses the x-axis at x = 1 and x = 3.

中文：分解得 y = (x – 1)(x – 3)，因此曲线与 x 轴交于 x = 1 和 x = 3。
English: The y-intercept is y = 3 when x = 0.

中文：当 x = 0 时，y 轴截距为 y = 3。
English: The axis of symmetry lies halfway between the roots, so x = 2. Substituting x = 2 gives y = 4 – 8 + 3 = -1. Thus the vertex is (2, -1).

中文：对称轴位于两个根的中点，所以 x = 2。代入 x = 2 得 y = 4 – 8 + 3 = -1。因此顶点为 (2, -1)。

Remember to label the x- and y-axes, the intercepts, and the vertex clearly in any exam sketch.

考试画图时，记得标注 x 轴、y 轴、交点以及顶点。

8. Solving Quadratic Inequalities | 二次不等式

Quadratic inequalities often appear in higher-level IGCSE questions. The graph of a quadratic is a useful visual guide for finding the correct solution set.

二次不等式常常出现在更高阶的IGCSE题目中。二次函数的图像是找出正确解集的直观工具。

Example: Solve x² – 5x + 6 > 0.

示例：解不等式 x² – 5x + 6 > 0。

English: First factorise x² – 5x + 6 = (x – 2)(x – 3). The critical values are x = 2 and x = 3.

中文：先分解 x² – 5x + 6 = (x – 2)(x – 3)。临界值为 x = 2 和 x = 3。

English: Since the coefficient of x² is positive, the parabola opens upwards and the expression is positive outside the interval between the roots.

中文：因为 x² 的系数为正，抛物线开口向上，表达式在两个根之外的区间取正值。

x < 2 or x > 3

English: For the inequality x² – 5x + 6 < 0, the solution is the interval between the roots, so 2 < x < 3.

中文：对于不等式 x² – 5x + 6 < 0，解集位于两根之间，即 2 < x < 3。

English: If the inequality symbol is ≤ or ≥, include the critical values using square brackets or a closed dot on a number line.

中文：如果不等号是 ≤ 或 ≥，需要把临界值包含在解集中，可以在数轴上用实心点表示。
English: Test one value in each interval to avoid sign errors.

中文：在每个区间内取一个测试值代入，可以避免符号错误。

9. Roots and the Vertex | 根与顶点

The roots of a quadratic equation correspond to the x-intercepts of its graph. The vertex gives the maximum or minimum value of the quadratic function. These ideas link algebra to geometry.

二次方程的根对应其图像与 x 轴的交点。顶点给出二次函数的最大值或最小值。这些想法把代数与几何联系了起来。

English: The completed square form y = a(x – h)² + k immediately gives the vertex (h, k).

中文：顶点式 y = a(x – h)² + k 直接给出顶点坐标 (h, k)。

Example: y = 2(x – 3)² – 8 has vertex (3, -8). Because a = 2 > 0, this is a minimum point.

例如：y = 2(x – 3)² – 8 的顶点为 (3, -8)。因为 a = 2 > 0，所以这是最低点。

English: Solving the equation a(x – h)² + k = 0 gives the x-intercepts. If k and a have opposite signs, the graph has two real roots.

中文：解方程 a(x – h)² + k = 0 可以得到 x 截距。如果 k 与 a 符号相反，则图像有两个实根。

Vertex x-coordinate = -b / 2a

English: This formula is very fast. Once you find the x-coordinate, substitute it into the quadratic expression to find the y-coordinate of the vertex.

中文：这个公式非常快捷。求出 x 坐标后，将其代入二次表达式即可得到顶点的 y 坐标。

10. Applications and Exam Strategies | 应用与考试策略

Quadratic equations are used in many real-world problems, including calculating areas, projectile motion, and solving geometric puzzles. In IGCSE exams, application questions usually require you to build a quadratic equation and then solve it.

二次方程广泛应用于现实问题，包括面积计算、抛体运动和几何问题。在IGCSE考试中，应用题通常需要你先建立二次方程，再求解。

Example: A rectangle has length (x + 4) m and width (x – 1) m. Its area is 54 m². Find x.

示例：一个矩形的长为 (x + 4) m，宽为 (x – 1) m，面积为 54 m²。求 x。

(x + 4)(x – 1) = 54 ⇒ x² + 3x – 4 = 54 ⇒ x² + 3x – 58 = 0

English: Use the quadratic formula with a = 1, b = 3, c = -58.

中文：使用公式法，此时 a = 1，b = 3，c = -58。

x = (-3 ± √(9 + 232)) / 2 = (-3 ± √241) / 2

Since x must be positive, choose the positive root. Reject the negative root because a length cannot be negative.

因为 x 必须为正，所以选择正的根。负根应舍去，因为长度不能为负数。

English: Always read the question carefully to see whether every root is acceptable in the context.

中文：做题时仔细审题，根据实际意义判断每个根是否可取。
English: Write down the formula, substitute numbers, and show each simplification step. Correct working earns method marks even if the final answer is wrong.

中文：写出公式、代入数值，并展示每一步化简。即使最终答案出错，正确的过程也能获得步骤分。
English: Check your solutions by substituting them back into the original equation. This simple step often catches calculation or sign errors.

中文：将解代回原方程进行检验。这个简单步骤常常能发现计算错误或符号错误。

By mastering these techniques, you will be able to solve quadratic problems quickly and accurately in the IGCSE exam.

掌握这些技巧后，你就能在IGCSE考试中快速准确地解决二次方程相关问题。

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Mastering Quadratic Equations for IGCSE Mathematics | 掌握二次方程：IGCSE数学核心专题