Solving Quadratic Equations by Factorisation | 因式分解法解一元二次方程

📚 Solving Quadratic Equations by Factorisation | 因式分解法解一元二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. Learning to solve it by factorisation develops algebraic fluency and appears frequently in algebra, graphs, and problem-solving questions.

一元二次方程是 IGCSE 数学中最重要的话题之一。学会用因式分解法求解二次方程可以提升代数运算能力，并且在代数、图像和应用题中经常出现。

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

A quadratic equation is any equation that can be rearranged into the standard form ax² + bx + c = 0, where x is the unknown and a, b, c are constants with a ≠ 0.

一元二次方程是任何可以整理成标准形式 ax² + bx + c = 0 的方程，其中 x 是未知数，a、b、c 是常数，且 a ≠ 0。

The word “quadratic” comes from the Latin word “quadratus”, meaning square, because the highest power of x is 2. If a = 0, the equation becomes linear, not quadratic.

“quadratic” 一词来源于拉丁语 “quadratus”，意为“平方”，因为 x 的最高次数是 2。如果 a = 0，方程就变成一次方程，而不再是二次方程。

ax² + bx + c = 0, a ≠ 0

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

In IGCSE exams, you should usually write a quadratic equation in standard form before trying to factorise it.

在 IGCSE 考试中，通常需要先把一元二次方程写成标准形式，再尝试因式分解。

The coefficient a is called the leading coefficient or quadratic coefficient. The coefficient b is the linear coefficient, and c is the constant term.

系数 a 称为二次项系数，系数 b 称为一次项系数，c 称为常数项。

For example, 2x² – 5x + 3 = 0 is in standard form, with a = 2, b = -5 and c = 3.

例如，2x² – 5x + 3 = 0 就是标准形式，其中 a = 2，b = -5，c = 3。

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3. The Idea Behind Factorisation | 因式分解的基本思想

To solve a quadratic equation by factorisation, we write the quadratic expression as a product of two linear factors.

用因式分解法解一元二次方程时，我们把二次式写成两个一次因式的乘积。

This method works because of the zero product property: if p × q = 0, then p = 0 or q = 0, or both.

这种方法的基础是零乘积性质：如果 p × q = 0，那么 p = 0 或 q = 0，或者两者都为零。

p × q = 0 → p = 0 or q = 0

Once the expression is factorised, the solutions come directly from setting each factor to zero.

一旦表达式被因式分解，解就可以直接由令每个因式为零得出。

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4. Factorising x² + bx + c | 分解 x² + bx + c 型三项式

When a = 1, look for two numbers that multiply to give c and add to give b.

当 a = 1 时，寻找两个数，使它们的乘积等于 c，和等于 b。

For example, to factorise x² + 7x + 10, we need two numbers with product 10 and sum 7. The numbers are 2 and 5, so x² + 7x + 10 = (x + 2)(x + 5).

例如，分解 x² + 7x + 10 时，需要两个数，乘积为 10，和为 7。这两个数是 2 和 5，因此 x² + 7x + 10 = (x + 2)(x + 5)。

If c is positive but b is negative, both numbers must be negative. If c is negative, the two numbers must have opposite signs.

如果 c 为正但 b 为负，那么两个数都为负；如果 c 为负，那么两个数的符号相反。

For x² – 8x + 15, the numbers -3 and -5 give product 15 and sum -8, so the factors are (x – 3)(x – 5).

对于 x² – 8x + 15，数 -3 和 -5 的乘积为 15，和为 -8，因此因式为 (x – 3)(x – 5)。

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5. Factorising ax² + bx + c (a > 1) | 分解 ax² + bx + c 型（a > 1）

When a > 1, use the ac method or trial and error. Multiply a by c, then find two numbers that multiply to ac and add to b.

当 a > 1 时，可以使用 ac 方法或试错法。先求 a × c，再找两个

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