Quadratic Equations: Factorisation, Completing the Square and the Quadratic Formula | IGCSE 数学二次方程全解

📚 Quadratic Equations: Factorisation, Completing the Square and the Quadratic Formula | IGCSE 数学二次方程全解

A quadratic equation is one of the most important algebraic skills in IGCSE Mathematics. It appears in Paper 2 and Paper 4, often linked to graphs, inequalities, and real-life modelling. This article covers the standard form, three solving methods, the discriminant, graph interpretation, and common exam pitfalls.

二次方程是 IGCSE 数学最重要的代数技能之一，经常出现在 Paper 2 和 Paper 4 中，并与图像、不等式和现实建模结合考查。本文涵盖标准形式、三种求解方法、判别式、图像解读以及常见考试陷阱。

1. Introduction to Quadratic Equations | 二次方程简介

+ c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear.

二次方程是次数为 2 的方程，即未知数 x 的最高次幂为 x²。其一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。如果 a = 0，方程就变成一次方程。

equations usually have two solutions, which may be real or non-real depending on the coefficients.

例如，2x² + 3x - 5 = 0 是二次方程，而 4x - 7 = 0 是一次方程。二次方程通常有两个解，依据系数的不同，解可能是实数，也可能不是实数。

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2. Standard Form and Coefficients | 标准形式与系数

-2 and c = 4. In x² + 5x = 6, first subtract 6 from both sides to obtain x² + 5x - 6 = 0.

求解前，务必先将方程整理成标准形式 ax² + bx + c = 0。仔细识别 a、b 和 c，包括符号。在 3x² - 2x + 4 = 0 中，a = 3，b = -2，c = 4。对于 x² + 5x = 6，应先将两边减 6，得到 x² + 5x - 6 = 0。

This standard form is essential because factorisation and the quadratic formula both depend on correct values of a, b and c. If the equation has brackets, expand and simplify first, then move all terms to one side.

标准形式非常关键，因为因式分解法和求根公式都依赖于正确的 a、b、c 值。如果方程含有括号，应先去括号并化简，再将所有项移到一边。

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3. Solving by Factorisation | 因式分解法

o numbers that multiply to 10 and add to 7: 2 and 5. So x² + 7x + 10 = (x + 2)(x + 5) = 0.

因式分解法适用于可将二次式写成两个一次括号乘积的情况。对于 x² + 7x + 10 = 0，找到两个数，乘积为 10、和为 7，即 2 和 5。因此 x² + 7x + 10 = (x + 2)(x + 5) = 0。

ual to zero: x + 2 = 0 gives x = -2; x + 5 = 0 gives x = -5. Always check by substitution.

令每个括号等于零：x + 2 = 0 得 x = -2；x + 5 = 0 得 x = -5。最后一定要代入原方程检验。

ddle term or use trial and error: 2x² + 5x + 3 = (2x + 3)(x + 1). Then x = -3/2 or x = -1.

当首项系数不是 1 时，例如 2x² + 5x + 3，可以拆分中间项或尝试括号：2x² + 5x + 3 = (2x + 3)(x + 1)，于是 x = -3/2 或 x = -1。

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4. Solving by Completing the Square | 配方法

Write x² + 6x = (x + 3)² - 9. Therefore x² + 6x + 4 = (x + 3)² - 9 + 4 = (x + 3)² - 5 = 0.

配方法将 x² + bx 转化为 (x + p)² + q 的形式。对于 x² + 6x + 4 = 0，取 6 的一半 3，平方得 9。写出 x² + 6x = (x + 3)² - 9。因此 x² + 6x + 4 = (x + 3)² - 9 + 4 = (x + 3)² - 5 = 0。

Then solve (x + 3)² = 5, so x + 3 = ±√5, giving x = -3 ± √5. This method is useful for finding the vertex of a graph and proving the quadratic formula.

然后解 (x + 3)² = 5，得 x + 3 = ±√5，所以 x = -3 ± √5。配方法在求图像顶点和推导求根公式时非常有用。

2x² + 8x + 5 = 0, first divide through by 2: x² + 4x + 5/2 = 0, then complete the square.

如果 x² 的系数不是 1，例如 2x² + 8x + 5 = 0，应先将整个方程除以 2，得到 x² + 4x + 5/2 = 0，再配方。

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5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation in standard form. It is:

x = [ -b ± √b² - 4ac ] / (2a)

求根公式可以求解任何标准形式的二次方程，公式如下：

x = [ -b ± √b² - 4ac ] / (2a)

0. The symbol ± means there are usually two solutions: one using plus and one using minus.

其中 a、b 和 c 是 ax² + bx + c = 0 中的系数。符号 ± 表示通常有两个解：一个用加号，另一个用减号。

For 2x² + 3x - 2 = 0, a = 2, b = 3, c = -2. Then x = [-3 ± √9 + 16] / 4 = [-3 ± 5] / 4, giving x = 1/2 or x = -2.

对于 2x² + 3x - 2 = 0，a = 2，b = 3，c = -2。则 x = [-3 ± √9 + 16] / 4 = [-3 ± 5] / 4，得 x = 1/2 或 x = -2。

This formula is especially useful when factorisation is difficult or when the roots are irrational. Remember to simplify the square root and the fraction fully if possible.

当因式分解较难或根为无理数时，求根公式尤其有用。记得尽可能化简根号和分数。

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

The expression b² - 4ac inside the square root is called the discriminant, often written as Δ. It tells you how many real roots the equation has without solving it.

根号内的表达式 b² - 4ac 称为判别式，通常记作 Δ。它可以在不求解的情况下判断方程有多少个实数根。

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

• 如果 Δ > 0，方程有两个不同的实数根。
• 如果 Δ = 0，方程有一个重根。
• 如果 Δ < 0，方程没有实数根。

s one repeated root x = 2. For x² + x + 1 = 0, Δ = 1 - 4 = -3, so there are no real roots.

对于 x² - 4x + 4 = 0，Δ = (-4)² - 4(1)(4) = 0，因此有一个重根 x = 2。对于 x² + x + 1 = 0，Δ = 1 - 4 = -3，所以没有实数根。

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7. Graphical Interpretation | 图像解读

ordinates where the parabola crosses the x-axis. If the graph touches the x-axis at one point, the discriminant is zero. If it never crosses, the discriminant is negative.

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