Quadratic Equations: Factorising, Completing the Square and the Formula | 二次方程：因式分解、配方法与公式法

📚 Quadratic Equations: Factorising, Completing the Square and the Formula | 二次方程：因式分解、配方法与公式法

ising, and interpret the roots in context. Extended candidates must also complete the square, use the quadratic formula, and link the discriminant to the number of real solutions.

二次方程是剑桥 IGCSE 数学大纲中最重要的主题之一。无论你参加核心课程还是扩展课程，都需要识别标准形式 ax² + bx + c = 0，通过因式分解解方程，并结合实际情境解释根的意义。扩展课程考生还必须掌握配方法、使用求根公式，并将判别式与实数解的个数联系起来。

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

stants and a ≠ 0. The highest power of the variable x is 2, which gives the equation its curved graph called a parabola.

二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数且 a ≠ 0。变量 x 的最高次数是 2，这使得它的图像是一条称为抛物线的曲线。

The expression ax² + bx + c is called a quadratic expression. If the equation is set equal to zero, the values of x that satisfy it are called the roots or solutions of the equation.

表达式 ax² + bx + c 称为二次式。如果令方程等于零，那么满足方程的 x 值称为方程的根或解。

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2. Core Method 1: Factorising x² + bx + c | 核心方法一：分解 x² + bx + c

0, we find 1 and 5 because 1 × 5 = 5 and 1 + 5 = 6. Thus x² + 6x + 5 = (x + 1)(x + 5) = 0.

当 a = 1 时，我们需要找到两个数，它们的乘积等于 c，和等于 b。例如，解 x² + 6x + 5 = 0 时，我们找到 1 和 5，因为 1 × 5 = 5 且 1 + 5 = 6。因此 x² + 6x + 5 = (x + 1)(x + 5) = 0。

0, giving x = −1 or x = −5. Always check by substituting back into the original equation.

因式分解后，令每个括号分别等于零：x + 1 = 0 或 x + 5 = 0，得到 x = −1 或 x = −5。一定要代回原方程进行检验。

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3. Core Method 2: Factorising ax² + bx + c | 核心方法二：分解 ax² + bx + c

When a ≠ 1, factorising is more involved. Multiply a and c to find ac, then find two numbers that multiply to ac and add to b. Split the middle term and factorise by grouping.

当 a ≠ 1 时，因式分解更复杂。先将 a 和 c 相乘得到 ac，然后找两个数，它们的乘积等于 ac，和等于 b。拆分中间项，再分组分解。

+ x + 3, group to get 2x(x + 3) + 1(x + 3), then (2x + 1)(x + 3) = 0. So x = −½ or x = −3.

例如：解 2x² + 7x + 3 = 0。我们需要两个数，乘积为 2 × 3 = 6，和为 7，因此是 6 和 1。重写为 2x² + 6x + x + 3，分组得到 2x(x + 3) + 1(x + 3)，然后 (2x + 1)(x + 3) = 0。所以 x = −½ 或 x = −3。

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4. Extended Method 3: Completing the Square | 扩展方法三：配方法

can be written as (x + 3)² − 4 because (x + 3)² = x² + 6x + 9 and we subtract 4 to get +5.

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。这有助于找到抛物线的顶点，并求解不容易因式分解的方程。例如，x² + 6x + 5 可以写成 (x + 3)² − 4，因为 (x + 3)² = x² + 6x + 9，再减去 4 得到 +5。

e square, set (x + 3)² − 4 = 0, then (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.

用配方法解 x² + 6x + 5 = 0，令 (x + 3)² − 4 = 0，然后 (x + 3)² = 4，所以 x + 3 = ±2，得到 x = −1 或 x = −5。

When the coefficient of x² is not 1, first factor it out from the x² and x terms before completing the square inside the bracket.

当 x² 的系数不是 1 时，先将其从 x² 项和 x 项中提出，再在括号内完成配方。

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5. Extended Method 4: The Quadratic Formula | 扩展方法四：公式法

The quadratic formula solves any quadratic equation ax² + bx + c = 0. It states that:

x = (−b ± √b² − 4ac) ÷ (2a)

求根公式可以解任何二次方程 ax² + bx + c = 0。公式为：

x = (−b ± √b² − 4ac) ÷ (2a)

The quadratic formula is derived by completing the square on the general quadratic equation. You can use it when factorising is difficult, when coefficients are decimals or fractions, or when a question specifically asks for answers correct to a given number of decimal places.

求根公式是由对一般二次方程进行配方法推导出来的。当因式分解困难、系数为小数或分数、或者题目明确要求将答案保留到指定小数位时，你都可以使用它。

uation 2x² + 7x + 3 = 0, a = 2, b = 7, c = 3. Substitute into the formula to get x = (−7 ± √49 − 24) ÷ 4 = (−7 ± √25) ÷ 4 = (−7 ± 5) ÷ 4, so x = −½ or x = −3.

对于方程 2x² + 7x + 3 = 0，a = 2，b = 7，c = 3。代入公式得到 x = (−7 ± √49 − 24) ÷ 4 = (−7 ± √25) ÷ 4 = (−7 ± 5) ÷ 4，所以 x = −½ 或 x = −3。

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6. The Discriminant and the Nature of Roots | 判别式与根的性质

Δ = b² − 4ac. It tells you how many real roots the equation has without solving it fully.

判别式是求根公式中平方根号下的部分：Δ = b² − 4ac。它告诉你在不完整求解的情况下方程有多少个实数根。

real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are no real roots.

如果 Δ > 0，有两个不同的实数根。如果 Δ = 0，有一个重复的实数根。如果 Δ < 0，没有实数根。

= 16. Since Δ > 0, there are two distinct real roots, which we already found as −1 and −5.

例如：对于 x² + 6x + 5 = 0，Δ = 6² − 4 × 1 × 5 = 36 − 20 = 16。因为 Δ > 0，所以有两个不同的实数根，我们已经求出为 −1 和 −5。

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7. Solving Quadratic Equations by Graphing | 用图像法解二次方程

dinates where the graph crosses the x-axis. These points are called x-intercepts or roots.

二次方程 y = ax² + bx + c 表示一条抛物线。方程 ax² + bx + c = 0 的解就是图像与 x 轴交点的横坐标。这些点称为 x 轴截距或根。

If the graph touches the x-axis at one point, the equation has one repeated root. If the graph does not cross the x-axis, the equation has no real roots. This visual link is often tested in both Core and Extended papers.

如果图像在一点与 x 轴相切，方程有一个重复根。如果图像不与 x 轴

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