Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与二次公式

📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与二次公式

In IGCSE Mathematics, quadratic equations appear in algebra, geometry and real-life modelling. A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a ≠ 0. This article explains the three main algebraic methods for solving quadratics: factorising, completing the square and using the quadratic formula. You will also learn how the discriminant controls the number of roots and how to sketch the related parabola.

在 IGCSE 数学中，二次方程广泛出现在代数、几何和实际建模中。二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a ≠ 0。本文讲解三种主要的代数解法：因式分解法、配方法和二次公式法。你还将学习判别式如何决定根的个数，以及如何绘制相应的抛物线。

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

A quadratic equation is a polynomial equation of degree 2. In standard form it is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were zero, the equation would become linear.

二次方程是次数为 2 的多项式方程。标准形式写作 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。如果 a 为零，方程就变成了一次方程。

The solutions of a quadratic equation are called roots. A quadratic equation can have two real roots, one repeated real root, or no real roots. This depends on the value of the discriminant b² − 4ac.

二次方程的解称为根。二次方程可能有两个实根、一个重根或没有实根。这取决于判别式 b² − 4ac 的值。

For example, x² − 5x + 6 = 0 is a quadratic equation with a = 1, b = −5 and c = 6.

例如，x² − 5x + 6 = 0 是一个二次方程，其中 a = 1、b = −5、c = 6。

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2. Solving by Factorising | 因式分解法

Factorising is usually the fastest method when the quadratic expression can be written as a product of two linear brackets. For a monic quadratic x² + bx + c, we look for two numbers that multiply to c and add to b.

当二次式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。对于首项系数为 1 的二次式 x² + bx + c，我们寻找两个数，使它们的乘积为 c，和为 b。

If the two numbers are p and q, then x² + bx + c = (x + p)(x + q). Setting each bracket equal to zero gives the roots x = −p and x = −q.

如果这两个数是 p 和 q，那么 x² + bx + c = (x + p)(x + q)。令每个括号等于零，就得到根 x = −p 和 x = −q。

For a non-monic quadratic ax² + bx + c, you may need to split the middle term or use trial and error. Always expand your brackets to check the factorisation.

对于首项系数不为 1 的二次式 ax² + bx + c，你可能需要拆分中间项或使用试错法。展开括号检查因式分解总是必要的。

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3. Worked Example: Factorising | 例题：因式分解

Solve x² − 7x + 12 = 0 by factorising.

用因式分解法解方程 x² − 7x + 12 = 0。

We need two numbers whose product is 12 and sum is −7. The numbers are −3 and −4 because (−3) × (−4) = 12 and (−3) + (−4) = −7.

我们需要两个数，乘积为 12，和为 −7。这两个数是 −3 和 −4，因为 (−3) × (−4) = 12，且 (−3) + (−4) = −7。

Therefore, x² − 7x + 12 = (x − 3)(x − 4) = 0.

因此，x² − 7x + 12 = (x − 3)(x − 4) = 0。

Setting each factor to zero gives x − 3 = 0 so x = 3, or x − 4 = 0 so x = 4.

令每个因式为零，得到 x − 3 = 0，所以 x = 3；或 x − 4 = 0，所以 x = 4。

The roots are x = 3 and x = 4.

根为 x = 3 和 x = 4。

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4. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This method is useful for solving equations, finding the vertex of a parabola and proving the quadratic formula.

配方法将二次式改写为 a(x + p)² + q 的形式。这种方法对解方程、求抛物线的顶点以及证明二次公式都很有用。

For x² + bx, add and subtract (b/2)² to create a perfect square: x² + bx = (x + b/2)² − (b/2)².

对于 x² + bx，加上并减去 (b/2)²，可构成完全平方：x² + bx = (x + b/2)² − (b/2)²。

For a general quadratic ax² + bx + c, first factor out a from the first two terms, then complete the square inside the bracket.

对于一般二次式 ax² + bx + c，先从第一、二项中提出 a，再在括号内完成配方。

When solving, rearrange the completed-square form to isolate x. Remember to include both positive and negative square roots.

解方程时，重新整理配方形式，把 x 分离出来。记得要同时取正、负平方根。

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5. Worked Example: Completing the Square | 例题：配方法

Solve x² + 6x + 2 = 0 by completing the square.

用配方法解方程 x² + 6x + 2 = 0。

Start with x² + 6x. Half of 6 is 3, and 3² = 9, so x² + 6x = (x + 3)² − 9.

从 x² + 6x 开始。6 的一半是 3，3² = 9，所以 x² + 6x = (x + 3)² − 9。

Substitute back: x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7 = 0.

代回原式：x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7 = 0。

Then (x + 3)² = 7, so x +

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