Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations appear in almost every IGCSE Mathematics paper, from Paper 2 to Paper 4, in both non-calculator and calculator sections. They are also the foundation for later topics such as functions, graphs and coordinate geometry. Understanding the three algebraic methods and the graphical interpretation of roots will give you a clear advantage in the examination.

二次方程几乎出现在每一份IGCSE数学试卷中，无论是Paper 2还是Paper 4，也无论是非计算器部分还是计算器部分。它更是函数、图像和坐标几何等后续专题的基础。掌握三种代数方法以及根的图像含义，能让你在考试中占据明显优势。

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

A quadratic equation is an equation of degree 2 in one variable, written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是含一个变量、最高次数为2的方程，一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

The condition a ≠ 0 is essential: if a = 0, the equation becomes linear and the x² term disappears. A quadratic equation can have at most two roots.

条件 a ≠ 0 至关重要：若 a = 0，方程就变成了一次方程，x² 项也随之消失。二次方程至多有两个根。

Typical examples include x² − 4x + 3 = 0, 2x² + 3x − 5 = 0 and x² = 9. Notice that the highest power of x is always 2.

典型的例子包括 x² − 4x + 3 = 0、2x² + 3x − 5 = 0 以及 x² = 9。注意 x 的最高次数始终是 2。

Solving a quadratic equation means finding all values of x that make the equation true. These values are called the roots or solutions of the equation.

解二次方程就是求出所有能使方程成立的 x 值，这些值称为方程的根或解。

2. Method 1: Factorisation | 方法一：因式分解

Factorisation is the quickest method when the quadratic has simple integer factors. For a monic quadratic x² + bx + c, look for two numbers whose product is c and whose sum is b.

当二次式具有简单的整数因式时，因式分解是最快捷的方法。对于首项系数为1的二次式 x² + bx + c，找出两个数，使它们乘积为 c、和为 b。

Example: solve x² − 5x + 6 = 0. We need two numbers with product 6 and sum −5; these are −2 and −3.

例：解 x² − 5x + 6 = 0。我们需要乘积为6、和为−5的两个数，它们是−2和−3。

x² − 5x + 6 = (x − 2)(x − 3) = 0

Using the zero product property, either x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

根据零乘积性质，x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

For non-monic quadratics such as 2x² + 7x + 3, use trial and error or split the middle term. Here 2x² + 7x + 3 = (2x + 1)(x + 3), giving roots x = −½ and x = −3.

对于形如 2x² + 7x + 3 的非首项系数二次式，可以尝试因式组合或拆中项。这里 2x² + 7x + 3 = (2x + 1)(x + 3)，所以根为 x = −½ 和 x = −3。

Always multiply your brackets back out to check that the factors are correct before writing the roots down.

在写下根之前，务必将括号重新展开，检查因式是否正确。

3. Method 2: Completing the Square | 方法二：配方法

Completing the square rewrites a quadratic in the form (x + p)² + q, which can then be solved by taking square roots.

配方法将二次式改写为 (x + p)² + q 的形式，然后通过开平方求解。

Starting from x² + bx + c = 0, add and subtract (b/2)² to form a perfect square.

从 x² + bx + c = 0 出发，加上并减去 (b/2)²，以构成完全平方。

Worked example: solve x² + 6x − 7 = 0. Here b = 6 so (b/2)² = 9.

例题：解 x² + 6x − 7 = 0。这里 b = 6，因此 (b/2)² = 9。

x² + 6x − 7 = (x + 3)² − 9 − 7 = 0

(x + 3)² = 16

x + 3 = ±4

Therefore x = 1 or x = −7. Note the ± sign when taking square roots: a positive number has two square roots.

因此 x = 1 或 x = −7。注意开平方时出现的 ± 号：一个正数有两个平方根。

This method also reveals the vertex of the parabola: y = (x + p)² + q has vertex (−p, q), which is useful for sketching graphs and for completing the square questions that do not require solving.

这种方法还能揭示抛物线的顶点：y = (x + p)² + q 的顶点为 (−p, q)，这对画图以及不需求解的配方法题目很有帮助。

4. Method 3: The Quadratic Formula | 方法三：求根公式

The quadratic formula works for every quadratic equation, even when factorisation is difficult or impossible.

求根公式适用于所有二次方程，即使因式分解很难或无法进行。

x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0

Worked example: solve 2x² + 5x − 3 = 0 using the formula. Here a = 2, b = 5 and c = −3. Substitute carefully.

例题：用求根公式解 2x² + 5x − 3 = 0。这里 a = 2，b = 5，c = −3。代入时要格外小心。

x = (−5 ± √(5² − 4 × 2 × (−3))) / (2 × 2) = (−5 ± √49) / 4

x = (−5 + 7) / 4 = 0.5 or x = (−5 − 7) / 4 = −3

So the roots are 0.5 and −3. Substitute each value back into the original equation to confirm both are correct.

因此根为 0.5 和 −3。将每个值代回原方程即可验证两者都正确。

On the non-calculator paper, leave answers in surd form such as (3 ± √5)/2 rather than rounding decimals. On the calculator paper, round to three significant figures unless told otherwise.

在非计算器试卷中，答案应以根式形式保留，如 (3 ± √5)/2，而不要写成近似小数。在计算器试卷中，除非另有说明，通常四舍五入到三位有效数字。

5. The Discriminant | 判别式

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

公式中的 b² − 4ac 称为判别式，通常用符号 Δ 表示。

The discriminant tells us the number and type of roots without actually solving the equation.

判别式可以在不解方程的情况下告诉我们根的个数和类型。

If Δ > 0, the equation has two distinct real roots.

若 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, the equation has one repeated real root.

若 Δ = 0，方程有一个重根（实数根）。
If Δ < 0, the equation has no real roots.

若 Δ < 0，方程没有实数根。

Example: x² + 4x + 5 = 0 has Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4 < 0, so it has no real roots.

例：x² + 4x + 5 = 0 的 Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4 < 0，因此没有实数根。

Graphically, the discriminant tells us whether the parabola cuts the x-axis twice, touches it once, or does not meet it at all.

从图像上看，判别式告诉我们抛物线是与 x 轴相交两次、相切一次，还是完全没有交点。

6. Solving

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Mastering Quadratic Equations | 掌握二次方程