Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most heavily tested topics in IGCSE Mathematics. Whether you are studying the Core or Extended syllabus, questions on factorising, solving, and sketching quadratics appear regularly. This guide breaks down every essential method step by step.

二次方程是 IGCSE 数学中考查最频繁的主题之一。无论你学习的是 Core（核心）还是 Extended（扩展）大纲，关于因式分解、求解和绘制二次函数图像的问题都经常出现。本指南将逐步拆解每一个关键方法。

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

A quadratic equation is any equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The coefficient a cannot be zero, otherwise the equation becomes linear.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。系数 a 不能为零，否则方程将退化为一次方程。

For example, 2x² − 3x + 1 = 0 is quadratic, while 2x − 3 = 0 is not.

例如，2x² − 3x + 1 = 0 是二次方程，而 2x − 3 = 0 不是。

It is important to always rearrange an equation into standard form before solving, because all solution methods assume this layout.

在求解前，务必先将方程整理成标准形式，因为所有解法都基于这一形式。

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2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic factorises neatly into two brackets.

当二次式能够顺利分解为两个括号相乘时，因式分解法是最快捷的方法。

Follow these steps:

请遵循以下步骤：

• Step 1: Rearrange into the form ax² + bx + c = 0.
• Step 2: Factorise the left-hand side into two linear factors.
• Step 3: Use the rule that if the product of two factors is zero, then at least one factor must be zero.
• Step 4: Solve each linear equation to find both values of x.

• 第一步：整理为 ax² + bx + c = 0 的形式。
• 第二步：将左侧分解为两个一次因式。
• 第三步：运用“两个因式相乘为零，则至少一个因式为零”的规则。
• 第四步：分别求解两个一次方程，得到 x 的两个值。

Example: Solve x² − 5x + 6 = 0.

例：求解 x² − 5x + 6 = 0。

We look for two numbers that multiply to give +6 and add to give −5: they are −2 and −3.

我们寻找两个相乘为 +6、相加为 −5 的数：它们是 −2 和 −3。

x² − 5x + 6 = (x − 2)(x − 3) = 0

Therefore 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。

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3. Solving by the Quadratic Formula | 公式法

When the quadratic does not factorise easily, the quadratic formula reliably solves any equation of the form ax² + bx + c = 0.

当二次式不易分解时，公式法可以可靠地求解任何 ax² + bx + c = 0 形式的方程。

x = (−b ± √(b² − 4ac)) / 2a

Example: Solve 2x² + 3x − 5 = 0 using the formula.

例：用公式法求解 2x² + 3x − 5 = 0。

Here a = 2, b = 3 and c = −5. Substituting:

这里 a = 2，b = 3，c = −5。代入得：

x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2) = (−3 ± √(9 + 40)) / 4 = (−3 ± √49) / 4

Since √49 = 7, x = (−3 + 7)/4 = 1 or x = (−3 − 7)/4 = −2.5.

因为 √49 = 7，所以 x = (−3 + 7)/4 = 1 或 x = (−3 − 7)/4 = −2.5。

Always substitute carefully, paying special attention to the value of c when it is negative.

代入时务必小心，当 c 为负数时尤其要注意符号。

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4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This is very useful for finding the vertex of a parabola and for solving equations.

配方法将二次式改写为 (x + p)² + q 的形式。这在求抛物线顶点和求解方程时非常有用。

Start by halving the coefficient of x.

首先将 x 的系数除以 2。

x² + bx = (x + b/2)² − (b/2)²

Example: Solve x² + 6x + 1 = 0 by completing the square.

例：用配方法求解 x² + 6x + 1 = 0。

(x + 3)² − 9 + 1 = 0 ⇒ (x + 3)² − 8 = 0

Then (x + 3)² = 8, so x + 3 = ±√8, giving x = −3 ± √8.

于是 (x + 3)² = 8，即 x + 3 = ±√8，所以 x = −3 ± √8。

If the coefficient of x² is not 1, factor it out first before completing the square.

如果 x² 的系数不为 1，请先提取该系数，再进行配方。

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5. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant, often written as Δ.

求根公式中的 b² − 4ac 称为判别式，通常记作 Δ。

Δ = b² − 4ac

The discriminant tells us how many real roots the equation has:

判别式告诉我们方程有多少个实数根：

Discriminant | 判别式	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 | 一个二重实数根
Δ < 0	No real roots | 无实数根

Example: For 2x² − 4x + 1 = 0, Δ = (−4)² − 4 × 2 × 1 = 16 − 8 = 8 > 0, so

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