IGCSE Mathematics: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像

📚 IGCSE Mathematics: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to real-world problems involving area, motion and finance. This article explains how to recognise, solve and sketch quadratics using several methods, with a focus on the skills examiners expect you to show.

二次方程贯穿 IGCSE 数学大纲，从代数、函数图像到涉及面积、运动与金融的实际问题。本文将讲解如何识别、求解和绘制二次函数图像，并重点介绍考试中需要展示的关键方法。

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

A quadratic equation in one variable is an equation that can be written in the standard form:

一元二次方程是可以写成以下标准形式的方程：

ax² + bx + c = 0, a ≠ 0

Here a, b and c are constants, and x is the unknown. The condition a ≠ 0 is essential because if a = 0, the equation becomes linear rather than quadratic.

其中 a、b、c 是常数，x 是未知数。条件 a ≠ 0 很重要，因为如果 a = 0，方程就变成一次方程而非二次方程。

The highest power of x is 2, which gives the equation its name ‘quadratic’. Common examples include x² – 5x + 6 = 0 and 2x² + 3x – 2 = 0. A quadratic equation may have two real solutions, one repeated solution, or no real solutions depending on its discriminant.

x 的最高次数是 2，因此称为二次方程。常见例子有 x² – 5x + 6 = 0 和 2x² + 3x – 2 = 0。根据判别式的不同，二次方程可能有两个实数解、一个重复实数解或没有实数解。

2. Solving by Factorisation | 因式分解法求解

Factorisation is often the fastest method when the quadratic expression can be written as a product of two linear factors. To use this method, first bring all terms to one side so the equation equals zero.

当二次式能写成两个一次因式的乘积时，因式分解通常是最快的方法。使用该方法时，先将所有项移到一边，使方程等于零。

For example, to solve x² – 5x + 6 = 0, find two numbers that multiply to +6 and add to -5. These are -2 and -3, so we can write:

例如，解 x² – 5x + 6 = 0 时，找出两个数，它们的乘积为 +6，和为 -5。这两个数是 -2 和 -3，因此可以写成：

x² – 5x + 6 = (x – 2)(x – 3) = 0

Set each factor equal to zero: x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3. Always check your solutions by substituting them back into the original equation.

令每个因式等于零：x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。始终要将解代回原方程进行检验。

If the coefficient of x² is not 1, for example 2x² + 3x – 2 = 0, use a trial method or split the middle term. Splitting the middle term gives 2x² + 4x – x – 2 = 0, then factor by grouping: 2x(x + 2) – 1(x + 2) = 0, so (2x – 1)(x + 2) = 0. Hence x = 1/2 or x = -2.

若 x² 的系数不是 1，例如 2x² + 3x – 2 = 0，可使用十字相乘法或拆项法。拆项后得到 2x² + 4x – x – 2 = 0，分组分解为 2x(x + 2) – 1(x + 2) = 0，因此 (2x – 1)(x + 2) = 0，解得 x = 1/2 或 x = -2。

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

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This form makes the vertex of the graph easier to find and is also useful for solving equations that do not factorise neatly.

配方法将二次式写成 a(x + p)² + q 的形式。这种形式便于求图像的顶点，也适用于不易因式分解的方程。

For x² + 6x + 5 = 0, take half of the coefficient of x, which is 3, and square it to get 9. Then write:

对于 x² + 6x + 5 = 0，取 x 系数的一半，即 3，再将其平方得到 9。然后写成：

x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4 = 0

Now solve (x + 3)² = 4, so x + 3 = ±√4 = ±2, giving x = -1 or x = -5. This method works even when the roots are irrational or complex.

接着解 (x + 3)² = 4，得 x + 3 = ±√4 = ±2，所以 x = -1 或 x = -5。即使根是无理数或复数，配方法仍然有效。

When the coefficient a is not 1, first factor it out from the x² and x terms before completing the square. For example, 2x² + 8x + 3 becomes 2(x² + 4x) + 3, then complete the square inside the bracket.

当系数 a 不为 1 时，先将 a 从 x² 项和 x 项中提出，再在括号内完成配方。例如 2x² + 8x + 3 可写成 2(x² + 4x) + 3，然后在括号内完成配方。

4. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation of the form ax² + bx + c = 0. It is a reliable method when factorisation is difficult or impossible.

二次公式可以求解任何形如 ax² + bx + c = 0 的二次方程。当因式分解困难或不可能时，它是可靠的方法。

x = (-b ± √(b² – 4ac)) / 2a

In this formula, a, b and c are the coefficients from the standard form. For example, to solve 2x² + 3x – 2 = 0, substitute a = 2, b = 3, c = -2:

公式中的 a、b、c 是标准形式中的系数。例如，解 2x² + 3x – 2 = 0 时，代入 a = 2, b = 3, c = -2：

x = (-3 ± √(3² – 4 × 2 × (-2))) / (2 × 2) = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4

This gives x = 2/4 = 1/2 or x = -8/4 = -2, matching the factorisation result. Always write the formula accurately and show your substitution clearly to gain method marks in exams.

由此得到 x = 2/4 = 1/2 或 x = -8/4 = -2，与因式分解的结果一致。考试中务必准确写出公式并清晰展示代入过程，以获得方法分。

5. The Discriminant | 判别式

The discriminant is the expression under the square root in the quadratic formula: b² – 4ac. It tells us how many real roots a quadratic equation has without actually solving it.

判别式是二次公式中根号下的表达式：b² – 4ac。它可以在不实际求解的情况下告诉我们二次方程有多少个实数根。

If b² – 4ac > 0, the equation has two distinct real roots, and the graph crosses the x-axis at two different points.

若 b² – 4ac > 0，方程有两个不同的实数根，图像与 x 轴在两个不同点相交。

If b² – 4ac = 0, the equation has one repeated real root, and the graph touches the x-axis at exactly one point, often called the turning point.

若 b² – 4ac = 0，方程有一个重复实数根，图像在恰好一个点处与 x 轴相切，该点通常称为转折点。

If b² – 4ac < 0, the equation has no real roots, and the graph does not intersect the x-axis at all.

若 b² – 4ac < 0，方程没有实数根，图像与 x 轴完全不相交。

For example, x² + 2x + 5 = 0 has discriminant 2² – 4 × 1 × 5 = 4 – 20 = -16, so it has no real solutions.

例如，x² + 2x + 5 = 0 的判别式为 2² – 4 × 1 × 5 = 4 – 20 = -16，因此它没有实数解。

6. Graphs of Quadratic Functions | 二次函数图像

A quadratic function of the form y = ax² + bx + c produces a curve called a parabola. The shape is symmetric and can open either upwards or downwards.

形如 y = ax² + bx + c 的二次函数图像是一条称为抛物线的曲线。该形状具有对称性，可以向上或向下开口。

If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The value of a also affects how wide or narrow the parabola is.

若 a > 0，抛物线开口向上，有最低点；若 a < 0，抛物线开口向下，有最高点。a 的值还会影响抛物线的宽窄程度。

The graph is symmetric about a vertical line passing through its vertex. This symmetry line is called the axis of symmetry and its equation is x = -b/2a.

图像关于经过顶点的一条竖直直线对称。这条对称线称为对称轴，其方程为 x = -b/2a。

7. Key Features of a Parabola | 抛物线的关键特征

When sketching a quadratic graph, the main features to identify are the y-intercept, the x-intercepts or roots, and the vertex.

绘制二次函数图像时，需要识别的主要特征有 y 轴截距、x 轴截距（或根）以及顶点。

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IGCSE Mathematics: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像