Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学：解二次方程

📚 Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学：解二次方程

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from basic factorising to real-life projectile problems. Mastering the three solution methods and understanding the discriminant will give you a strong advantage in both Core and Extended papers.

二次方程贯穿 IGCSE 数学课程，从基础因式分解到现实中的抛体运动问题。掌握三种解法并理解判别式，能让你在 Core 和 Extended 试卷中占据明显优势。

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

A quadratic equation is an equation of degree 2, meaning the highest power of the variable is 2. Its general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是次数为 2 的方程，即变量的最高次数是 2。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。

ax² + bx + c = 0, a ≠ 0

For example, x² – 5x + 6 = 0, 3x² + 2x – 1 = 0, and -x² + 4 = 0 are all quadratic equations. If a = 0, the equation becomes linear.

例如，x² – 5x + 6 = 0、3x² + 2x – 1 = 0 和 -x² + 4 = 0 都是二次方程。如果 a = 0，方程就变成了一次方程。

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2. Standard Form and Key Terminology | 标准形式与关键术语

Before solving a quadratic equation, always rearrange it into standard form: ax² + bx + c = 0. This means collecting all terms on one side and setting the other side equal to zero.

在解二次方程之前，一定要先把它整理为标准形式：ax² + bx + c = 0。也就是把所有项移到一边，让另一边等于零。

The coefficient a is the number in front of x², the coefficient b is the number in front of x, and c is the constant term. The solutions are called roots or x-intercepts.

系数 a 是 x² 前面的数，系数 b 是 x 前面的数，c 是常数项。方程的解称为根或 x 轴截距。

For example, 2x² = 5x – 3 should first be written as 2x² – 5x + 3 = 0. In this form, a = 2, b = -5 and c = 3.

例如，2x² = 5x – 3 应首先写成 2x² – 5x + 3 = 0。在这种形式下，a = 2，b = -5，c = 3。

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3. Solving by Factorising | 因式分解法

Factorising is often the fastest method when the quadratic has integer roots. Write the left-hand side as a product of two brackets, then set each bracket equal to zero.

当二次方程有整数根时，因式分解通常是最快的方法。把左边写成两个括号的乘积，然后令每个括号等于零。

For x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5: these are 2 and 3. So the equation becomes (x + 2)(x + 3) = 0.

对于 x² + 5x + 6 = 0，找到两个数，使它们相乘得 6、相加得 5：这两个数是 2 和 3。因此方程变为 (x + 2)(x + 3) = 0。

This gives x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3. Always check by substituting back into the original equation.

由此得到 x + 2 = 0 或 x + 3 = 0，所以 x = -2 或 x = -3。一定要代回原方程进行检验。

When a ≠ 1, factorising may require trial and error. For example, 2x² + 7x + 3 = 0 factors as (2x + 1)(x + 3) = 0, giving x = -½ or x = -3.

当 a ≠ 1 时，因式分解可能需要尝试。例如，2x² + 7x + 3 = 0 可分解为 (2x + 1)(x + 3) = 0，得到 x = -½ 或 x = -3。

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4. Solving by the Quadratic Formula | 二次公式法

The quadratic formula works for any quadratic equation, even when factorising is difficult or impossible. It is given by the formula below.

二次公式适用于任何二次方程，即使因式分解困难或不可能时也能使用。它由以下公式给出。

x = (-b ± √(b² – 4ac)) ÷ 2a

First identify a, b and c from standard form, then substitute them carefully. Calculate the discriminant b² – 4ac before taking the square root.

首先从标准形式中确定 a、b 和 c，然后仔细代入。在开方之前先计算判别式 b² – 4ac。

For x² – 3x – 10 = 0, a = 1, b = -3 and c = -10. Substituting gives x = (3 ± √(9 + 40)) ÷ 2 = (3 ± √49) ÷ 2, so x = (3 + 7) ÷ 2 = 5 or x = (3 – 7) ÷ 2 = -2.

对于 x² – 3x – 10 = 0，a = 1，b = -3，c = -10。代入得 x = (3 ± √(9 + 40)) ÷ 2 = (3 ± √49) ÷ 2，因此 x = (3 + 7) ÷ 2 = 5 或 x = (3 – 7) ÷ 2 = -2。

Be careful with negative signs: -(-3) = 3, and b² is (-3)² = 9. This is a common source of avoidable errors.

注意负号：-(-3) = 3，而 b² 是 (-3)² = 9。这是常见的可避免错误来源。

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5. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic in the form a(x – h)² + k = 0, which is useful for finding the vertex and for deriving the quadratic formula.

配方法把二次方程改写为 a(x – h)² + k = 0 的形式，这对求顶点和推导二次公式都很有用。

For x² + 6x + 5 = 0, first move the constant: x² + 6x = -5. Then add (6/2)² = 9 to both sides: x² + 6x + 9 = 4, which is (x + 3)² = 4.

对于 x² + 6x + 5 = 0，先把常数移项：x² + 6x = -5。然后两边加上 (6/2)² = 9：x² + 6x + 9 = 4，即 (x + 3)² = 4。

Taking square roots gives x + 3 = ±2, so x = -1 or x = -5. This method is especially important when the roots are not integers.

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