Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in almost every exam paper, often in multiple sections. Mastering the different methods of solving quadratics will boost both your confidence and your grade.

二次方程是 IGCSE 数学中最重要的话题之一。几乎每份试卷都会考到，甚至多次出现。掌握解二次方程的不同方法，会大大提升你的信心和成绩。

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

A quadratic equation is a polynomial equation of degree 2. This means the highest power of the unknown variable is 2. For example, x² = 9 and 2x² − 5x + 3 = 0 are both quadratic equations.

二次方程是次数为 2 的多项式方程，也就是说未知数的最高次数是 2。例如，x² = 9 和 2x² − 5x + 3 = 0 都是二次方程。

A key feature is that a quadratic equation can have zero, one, or two real solutions. These solutions are also called the roots of the equation.

二次方程的一个关键特征是它可能有零个、一个或两个实数解。这些解也叫做方程的根。

2. The Standard Form | 标准形式

Every quadratic equation can be written in the standard form:

每一个二次方程都可以写成标准形式：

ax² + bx + c = 0, where a ≠ 0

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The condition a ≠ 0 is essential because if a = 0, the equation would no longer be quadratic.

其中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。条件 a ≠ 0 至关重要，因为如果 a = 0，方程就不再是二次方程了。

For example, in the equation 3x² − 2x + 7 = 0, we have a = 3, b = −2 and c = 7.

例如，在方程 3x² − 2x + 7 = 0 中，a = 3，b = −2，c = 7。

3. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic has simple integer roots. To solve by factorisation, you rewrite the quadratic as a product of two brackets set equal to zero.

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。用因式分解求解时，你需要把二次式写成两个括号相乘的形式，并令其等于零。

Consider x² − 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give −5. Those numbers are −2 and −3.

考虑 x² − 5x + 6 = 0。我们需要找两个数，相乘等于 6，相加等于 −5。这两个数是 −2 和 −3。

(x − 2)(x − 3) = 0

If the product of two factors is zero, then at least one factor must be zero. So x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

如果两个因式的乘积为零，那么至少有一个因式为零。所以 x − 2 = 0 或 x − 3 = 0，得到 x = 2 或 x = 3。

When the coefficient of x² is not 1, you may need to factor by grouping or use trial and error. For example, 2x² + 5x + 3 = 0 factors as (2x + 3)(x + 1) = 0, giving x = −1.5 or x = −1.

当 x² 的系数不是 1 时，你可能需要分组分解或试错。例如，2x² + 5x + 3 = 0 分解为 (2x + 3)(x + 1) = 0，得到 x = −1.5 或 x = −1。

4. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that works for any quadratic equation. It also helps you find the turning point of a quadratic graph.

配方法是一种强大的技巧，适用于任何二次方程。它还能帮助你找到二次函数图像的顶点。

To complete the square for x² + 6x + 2 = 0, first halve the coefficient of x: (6/2)² = 9. Then rewrite:

要对 x² + 6x + 2 = 0 配方，首先取 x 系数的一半：(6/2)² = 9。然后改写：

(x + 3)² − 9 + 2 = 0 → (x + 3)² = 7

Now take the square root of both sides. Remember to include the ± sign:

现在对两边开平方。记得加上 ± 符号：

x + 3 = ±√7 → x = −3 ± √7

So the two solutions are x = −3 + √7 and x = −3 − √7. When the coefficient of x² is not 1, divide the whole equation by that coefficient first.

因此两个解是 x = −3 + √7 和 x = −3 − √7。当 x² 的系数不是 1 时，先把整个方程除以该系数。

5. The Quadratic Formula | 求根公式

The quadratic formula is a universal method that solves any quadratic equation. It is derived from completing the square, but you can use it directly.

求根公式是一种通用方法，可以解任何二次方程。它由配方法推导而来，但你可以直接使用。

x = (−b ± √(b² − 4ac)) / 2a

For the equation 2x² − 4x − 3 = 0, identify a = 2, b = −4, c = −3. Substitute into the formula:

对于方程 2x² − 4x − 3 = 0，确定 a = 2，b = −4，c = −3。代入公式：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

This gives x = (4 − √40)/4 ≈ −0.58 and x = (4 + √40)/4 ≈ 2.58. Always simplify the surd where possible.

得到 x = (4 − √40)/4 ≈ −0.58 和 x = (4 + √40)/4 ≈ 2.58。尽量化简根式。

In the exam, you should memorise the formula. Write it at the top of your working paper as soon as the exam begins, if allowed.

考试中，你应该记住这个公式。如果允许，考试一开始就把它写在草稿纸顶部。

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

The expression b² − 4ac is called the discriminant. It tells us how many real roots a quadratic equation has without solving it fully.

表达式 b² − 4ac 叫做判别式。无需完整求解，它就能告诉我们二次方程有几个实数根。

If b² − 4ac > 0, the equation has two distinct real roots.

如果 b² − 4ac > 0，方程有两个不相等的实数根。
If b² − 4ac = 0, the equation has exactly one real root (a repeated root).

如果 b² − 4ac = 0，方程只有一个实数根（重根）。
If b² − 4ac < 0, the equation has no real roots.

如果 b² − 4ac < 0，方程没有实数根。

For example, x² + 2x + 5 = 0 has discriminant 2² − 4(1)(5) =

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Solving Quadratic Equations | 解二次方程