Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear throughout the syllabus, from simple factorisation to complex word problems. In this article, we will study how to solve quadratic equations using four different methods, understand the discriminant, and avoid common mistakes.

一元二次方程是 IGCSE 数学中最重要的考点之一。从简单的因式分解到复杂的应用题，它们贯穿整个课程大纲。在本文中，我们将学习用四种不同的方法解一元二次方程，理解判别式的含义，并避开常见错误。

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1. What Is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is a polynomial equation of degree 2. The general form is ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. If a = 0, the equation becomes linear.

一元二次方程是次数为 2 的多项式方程。它的标准形式是 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。如果 a = 0，方程就退化为一次方程。

Examples of quadratic equations include x² + 5x + 6 = 0 and 2x² – 3x – 2 = 0. The highest power of the variable is always 2.

例如，x² + 5x + 6 = 0 和 2x² – 3x – 2 = 0 都是一元二次方程。变量的最高次数永远是 2。

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2. The Standard Form | 标准形式

Before solving any quadratic equation, always rearrange it into the standard form ax² + bx + c = 0. This means moving all terms to one side of the equals sign, leaving 0 on the other side.

在解任何一元二次方程之前，务必将它整理成标准形式 ax² + bx + c = 0。也就是说，把所有的项移到等号的一侧，让另一侧为 0。

For example, the equation 2x² + 5 = 3x can be rewritten as 2x² – 3x + 5 = 0 by subtracting 3x from both sides. Careful: the coefficient signs must be handled correctly in this step.

例如，方程 2x² + 5 = 3x 可以通过两边同时减去 3x 改写成 2x² – 3x + 5 = 0。注意：这一步中系数符号必须正确处理。

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3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when it works. We look for two numbers whose product equals ac and whose sum equals b. Then we split the middle term, factorise by grouping, and apply the zero product property.

因式分解法是可用时最快的方法。我们要寻找两个数，使它们的乘积等于 ac，和等于 b。然后拆分中间项、分组因式分解，并运用零乘积性质。

Consider x² + 7x + 12 = 0. We need two numbers whose product is 12 and whose sum is 7. These numbers are 3 and 4. Hence, x² + 7x + 12 = (x + 3)(x + 4) = 0.

以 x² + 7x + 12 = 0 为例。我们需要找到乘积为 12、和为 7 的两个数。这两个数就是 3 和 4。因此，x² + 7x + 12 = (x + 3)(x + 4) = 0。

When the coefficient a is not 1, the method still works, but we first multiply a by c. For 2x² + 5x + 3 = 0, ac = 6. Two numbers with product 6 and sum 5 are 2 and 3. Split the middle term: 2x² + 2x + 3x + 3 = 0, then factorise by grouping: 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1) = 0.

当二次项系数 a 不为 1 时，该方法依然适用，但需要先将 a 乘以 c。对于 2x² + 5x + 3 = 0，ac = 6。乘积为 6、和为 5 的两个数是 2 和 3。拆分中间项：2x² + 2x + 3x + 3 = 0，然后分组因式分解：2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1) = 0。

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4. The Zero Product Property | 零乘积性质

If the product of two factors is zero, then at least one of the factors must be zero. This simple property is the key to solving factorised equations.

如果两个因式的乘积为零，那么至少有一个因式必须为零。这个简单的性质是解因式分解后方程的关键。

From (x + 3)(x + 4) = 0, we write x + 3 = 0 or x + 4 = 0. Therefore, x = -3 or x = -4.

由 (x + 3)(x + 4) = 0，我们得到 x + 3 = 0 或 x + 4 = 0。因此，x = -3 或 x = -4。

A quadratic equation always has two roots, though sometimes they may turn out to be equal. Below is a quick comparison of two cases.

一元二次方程总是有两个根，尽管有时它们可能相等。下面是对两种情况的快速对比。

Equation 方程	Factorised Form 因式分解形式	Roots 根
x² – 9 = 0	(x – 3)(x + 3) = 0	x = 3, x = -3
x² – 6x + 9 = 0	(x – 3)² = 0	x = 3 (twice 双重根)

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5. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² = q. This method works for every quadratic equation, even when factorisation fails.

配方法将一元二次方程改写为 (x + p)² = q 的形式。该方法适用于所有一元二次方程，即使在因式分解行不通时也能使用。

Take x² + 6x – 7 = 0. First, move the constant to the other side: x² + 6x = 7. Then take half of 6, square it (3² = 9), and add it to both sides: x² + 6x + 9 = 16.

以 x² + 6x – 7 = 0 为例。首先，将常数项移到另一边：x² + 6x = 7。然后取 6 的一半，即 3，平方得 9，两边同时加上 9：x² + 6x + 9 = 16。

Now the left-hand side is a perfect square: (x + 3)² = 16. Taking square roots gives x + 3 = ±4, so x = 1 or x = -7.

此时左边就是完全平方式：(x + 3)² = 16。两边开平方得到 x + 3 = ±4，所以 x = 1 或 x = -7。

(x + p)² = q ⇔ x = -p ± √q

Remember: when taking square roots, you must include both the positive and negative roots.

记住：开平方时，必须包含正负两个根。

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6. The Quadratic Formula | 求根公式

For any quadratic equation ax² + bx + c = 0, the roots are given by the quadratic formula. This is the most reliable method and always works.

对于任意一元二次方程 ax² + bx + c = 0，求根公式给出了根的表达。这是最可靠的方法，永远适用。

x = (-b ± √(b² – 4ac)) / 2a

Let us solve 2x² – 4x – 3 = 0 using the formula. Here a = 2, b = -4, c = -3. Substituting gives:

让我们用公式解 2x² – 4x – 3 = 0。这里 a = 2，b = -4，c = -3。代入得：

x = (4 ± √((-4)² – 4 × 2 × (-3))) / (2 × 2) = (4 ± √40) / 4

Since √40 = 2√10, we obtain x = (4 ± 2√10) / 4 = (2 ± √10) / 2. Always simplify the surd and cancel common factors where possible.

因为 √40 = 2√10，我们得到 x = (4 ± 2√10) / 4 = (2 ± √10) / 2。务必化简根式，并在可能的情况下约去公因数。

Some IGCSE calculators can solve quadratic equations

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