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

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

A quadratic equation is one of the most important algebraic skills in IGCSE Mathematics. It appears in pure algebra, coordinate geometry, measurement and word problems. This revision guide covers the three main solving strategies - factorising, completing the square and the quadratic formula - as well as the discriminant and graphical interpretation.

二次方程是 IGCSE 数学中最重要的代数技能之一。它出现在纯代数、坐标几何、测量和应用题中。本复习指南涵盖三种主要解法——因式分解、配方法和求根公式，以及判别式和图像解释。

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1. Recognising a Quadratic Equation | 识别二次方程

A quadratic equation is an equation in which the highest power of the variable is 2. It usually involves a single variable, frequently written as x, and its graph is a parabola. The general shape of a quadratic expression is ax² + bx + c, where a, b and c are constants and a cannot be zero.

二次方程是变量最高次数为 2 的方程。它通常只含一个变量，一般写作 x，其图像是一条抛物线。二次表达式的一般形式为 ax² + bx + c，其中 a、b、c 为常数且 a 不能为零。

In IGCSE questions, a quadratic may be hidden inside a word problem about areas, projectile motion, profit or geometry. Identifying the highest power of the variable quickly tells you that a quadratic method is needed.

在 IGCSE 题目中，二次方程可能隐藏在关于面积、抛体运动、利润或几何的文字题中。快速识别变量的最高次数可以告诉你需要使用二次方程的求解方法。

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2. Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

and c are constants and a ≠ 0. For example, 3x² − 7x + 2 = 0 has a = 3, b = −7 and c = 2.

求解前，务必先将方程改写为标准形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。例如，3x² − 7x + 2 = 0 中 a = 3，b = −7，c = 2。

ackets first, such as (x + 2)(x − 3) = 4 becoming x² − x − 6 = 4 and then x² − x − 10 = 0.

如果方程不是标准形式，应把所有项移到同一边并化简。你可能需要先展开括号，例如 (x + 2)(x − 3) = 4 变为 x² − x − 6 = 4，再化为 x² − x − 10 = 0。

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3. Solving by Factorising | 因式分解法求解

im is to write ax² + bx + c = 0 as (px + q)(rx + s) = 0, where p, q, r and s are integers.

当二次方程有简单的整数根时，因式分解通常是最快的方法。目标是将 ax² + bx + c = 0 写成 (px + q)(rx + s) = 0，其中 p、q、r、s 为整数。

and multiply to 6. Always expand your brackets mentally to check the middle term before writing the final factors.

例如，x² + 5x + 6 = 0 可因式分解为 (x + 2)(x + 3) = 0，因为 2 和 3 相加为 5，相乘为 6。在写出最终因式前，务必在脑中展开括号以检查中间项是否正确。

+ 1)(x + 3). Checking gives 2x × 3 + 1 × x = 6x + x = 7x, so the factorisation is correct.

当 a 不等于 1 时，尝试不同的组合，例如 2x² + 7x + 3 = (2x + 1)(x + 3)。检验可得 2x × 3 + 1 × x = 6x + x = 7x，因此因式分解正确。

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4. The Zero Product Property | 零乘积性质

Once a quadratic is factorised, we use the zero product property: if two factors multiply to zero, then at least one factor must be zero. This gives us two separate linear equations that are easy to solve.

二次方程因式分解后，我们使用零乘积性质：如果两个因式的乘积为零，那么至少有一个因式为零。由此得到两个独立的一次方程，很容易求解。

l to state both solutions unless the context of the question restricts the answer, for example if x represents a length.

由 (x + 2)(x + 3) = 0，令 x + 2 = 0 或 x + 3 = 0，得到 x = −2 或 x = −3。除非题目的情境限制了答案，例如 x 表示长度，否则必须写出两个解。

This property only works when one side of the equation is zero. That is why rearranging into standard form is a crucial first step.

该性质仅在方程一边为零时成立。这就是为什么将方程整理为标准形式是关键的第一步。

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5. Completing the Square | 配方法

is useful when factorising is difficult, and it also helps find the vertex of a parabola.

配方法将二次方程改写为 (x + p)² + q = 0 的形式。这在因式分解困难时很有用，也有助于求抛物线的顶点。

0, half of 6 is 3, so we write (x + 3)² − 9 + 5 = 0, which simplifies to (x + 3)² − 4 = 0.

关键步骤是取 x 项系数的一半，平方后加上再减去该平方数。对于 x² + 6x + 5 = 0，6 的一半是 3，因此写成 (x + 3)² − 9 + 5 = 0，化简为 (x + 3)² − 4 = 0。

before completing the square, such as 2x² + 8x + 6 = 2(x² + 4x) + 6 = 2[(x + 2)² − 4] + 6.

接着求解得到 x + 3 = ±2，所以 x = −1 或 x = −5。当 x² 的系数不是 1 时，应先将该系数提出，再进行配方，例如 2x² + 8x + 6 = 2(x² + 4x) + 6 = 2[(x + 2)² − 4] + 6。

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6. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation in standard form. It is derived from completing the square and reads:

求根公式可以求解任何标准形式的二次方程。它由配方法推导而来，公式如下：

x = (−b ± √b² − 4ac) ÷ 2a

cially when they are negative. For 2x² + 3x − 2 = 0, a = 2, b = 3 and c = −2, so x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4, giving x = ½ or x = −2.

仔细代入 a、b、c 的值，尤其当它们为负数时。对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = −2，所以 x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4，得到 x = ½ 或 x = −2。

The formula is indispensable when factorising is not obvious or when coefficients are irrational. Always write the formula first to earn method marks, even if you make a substitution error later.

当因式分解不明显或系数为无理数时，求根公式不可或缺。始终先写出公式，以获得方法分，即使之后代入出错也能拿到部分分数。

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7. The Discriminant | 判别式

c. It tells you how many real solutions a quadratic has without requiring you to solve it.

判别式是求根公式中平方根号下的表达式：Δ = b² − 4ac。它告诉你在不求解的情况下二次方程有多少个实数解。

l root. If Δ < 0, there are no real roots, but two complex roots if you study further mathematics.

如果 Δ > 0，则有两个不同的实数根；如果 Δ = 0，则有一个重根；如果 Δ < 0，则没有实数根，但在进阶数学中会有两个复数根。

4x² − 12x + 9 = 0: Δ = (−12)² − 4 × 4 × 9 = 144 − 144 = 0, so there is one repeated root.

IGCSE 考题可能要求你不解方程就判断根的个数，例如计算 4x² − 12x + 9 = 0 的判别式：Δ = (−12)² − 4 × 4 × 9 = 144 − 144 = 0，因此有一个重根。

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8. Graphical Interpretation | 图像解释

solutions of ax² + bx + c = 0 are the x-coordinates where the parabola crosses the x-axis.

y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解就是抛物线与 x 轴交点的 x 坐标。

If the parabola touches the x-axis at one point, the equation has one repeated root. If it does not cross the x-axis, there are no real roots

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