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 frequently tested topics in IGCSE Mathematics. Whether you are aiming for a grade C or a grade A*, you must be fluent in the three standard solution methods: factorisation, completing the square and the quadratic formula.

二次方程是IGCSE数学中考查频率最高的考点之一。无论你的目标是C还是A*，都必须熟练掌握三种标准解法：因式分解法、配方法和求根公式。

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

A quadratic equation is an equation that can be written in the standard form shown below, where a, b and c are constants and a ≠ 0.

二次方程是指可以写成下面标准形式的方程，其中 a、b、c 为常数，且 a ≠ 0。

ax² + bx + c = 0

The highest power of x is 2, which is why the graph of the equation y = ax² + bx + c is a curve called a parabola. If a = 0, the x² term disappears and the equation becomes linear, so it is no longer quadratic.

x 的最高次数为 2，因此方程 y = ax² + bx + c 的图像是一条称为抛物线的曲线。如果 a = 0，x² 项消失，方程退化为一次方程，就不再是二次方程了。

For example, x² + 3x − 10 = 0 is quadratic, but x + 3 = 0 is linear.

例如，x² + 3x − 10 = 0 是二次方程，而 x + 3 = 0 是一次方程。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the expression factorises nicely. The idea is based on the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

当表达式能够顺利分解时，因式分解法是最快捷的方法。其依据是零乘积性质：如果两个表达式的乘积为零，那么其中至少有一个为零。

If AB = 0, then A = 0 or B = 0.

Follow these three steps. Step 1: rearrange the equation so that it is in the form ax² + bx + c = 0. Step 2: factorise the left-hand side. Step 3: set each factor equal to zero and solve the resulting linear equations.

按以下三步操作：第一步，把方程整理成 ax² + bx + c = 0 的形式；第二步，对左边进行因式分解；第三步，令每个因式分别为零，并解出相应的一次方程。

Example: solve x² − 5x + 6 = 0.

例：解方程 x² − 5x + 6 = 0。

x² − 5x + 6 = (x − 2)(x − 3) = 0

Therefore x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3. Always write the final answer as ‘x = 2 or x = 3’, not ‘x = 2 and x = 3’.

因此 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。最终答案应写成 x = 2 或 x = 3，而不是 x = 2 且 x = 3。

Special case: the difference of two squares. For example, x² − 9 = (x − 3)(x + 3) = 0 gives x = 3 or x = −3.

特殊情况：平方差公式。例如，x² − 9 = (x − 3)(x + 3) = 0，解得 x = 3 或 x = −3。

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