Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics, appearing in almost every exam paper. Mastering this topic requires understanding factorisation, the quadratic formula, completing the square, and graph drawing.

二次方程是 IGCSE 数学中最重要的话题之一，几乎出现在每份试卷中。掌握这一主题需要理解因式分解、二次公式、配方法以及图像绘制。

1. The General Form of a Quadratic Equation | 二次方程的一般形式

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

二次方程是指最高次项为 2 次的方程，即变量的最高次数为 2。其标准形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. For example, 2x² − 5x + 3 = 0 is a quadratic equation with a = 2, b = −5 and c = 3.

其中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。例如，2x² − 5x + 3 = 0 是一个二次方程，其中 a = 2，b = −5，c = 3。

A quadratic equation can have at most two solutions, known as roots. These roots may be real and distinct, real and equal, or complex, depending on the value of the discriminant.

二次方程最多有两个解，称为根。这些根可能是两个不相等的实数、两个相等的实数，或是复数，具体取决于判别式的值。

2. Expanding and Simplifying Quadratic Expressions | 展开与化简二次表达式

Before solving quadratics, you must be confident in expanding expressions. The distributive law states that a(b + c) = ab + ac, and for two bracket pairs, (x + m)(x + n) = x² + (m + n)x + mn.

在解二次方程之前，你必须熟练掌握展开表达式。乘法分配律指出 a(b + c) = ab + ac，而两个括号相乘时，(x + m)(x + n) = x² + (m + n)x + mn。

For example, expand (x + 3)(x − 5): multiply x by x to get x², then x by −5 to get −5x, then 3 by x to get 3x, and finally 3 by −5 to get −15. Collecting like terms gives x² − 2x − 15.

例如，展开 (x + 3)(x − 5)：x 乘以 x 得 x²，x 乘以 −5 得 −5x，3 乘以 x 得 3x，最后 3 乘以 −5 得 −15。合并同类项得到 x² − 2x − 15。

The mnemonic FOIL (First, Outer, Inner, Last) helps remember the order of multiplication. This skill is essential because it helps you check whether a factorisation is correct.

助记法 FOIL（First 首项，Outer 外项，Inner 内项，Last 末项）有助于记住相乘的顺序。这一技能至关重要，因为它能帮助你检查因式分解是否正确。

3. Factorising Quadratic Expressions | 二次表达式的因式分解

Factorising is the reverse process of expanding. When a = 1, look for two numbers whose product is c and whose sum is b. For x² + 7x + 12, the numbers 3 and 4 satisfy 3 × 4 = 12 and 3 + 4 = 7, so the factorised form is (x + 3)(x + 4).

因式分解是展开的逆过程。当 a = 1 时，寻找两个数，它们的乘积为 c，和为 b。对于 x² + 7x + 12，数字 3 和 4 满足 3 × 4 = 12 且 3 + 4 = 7，因此因式分解形式为 (x + 3)(x + 4)。

When a ≠ 1, such as 2x² + 7x + 3, we use the method of splitting the middle term. Multiply a and c (2 × 3 = 6), then find two numbers that multiply to 6 and add to 7; these are 1 and 6. Rewrite 7x as x + 6x and factorise by grouping.

当 a ≠ 1 时，例如 2x² + 7x + 3，我们使用拆中项法。将 a 和 c 相乘（2 × 3 = 6），然后找到两个数相乘得 6 且相加得 7；这两个数是 1 和 6。将 7x 改写为 x + 6x，再分组因式分解。

The steps are:

步骤如下：

• Multiply a and c to get ac.
• 将 a 和 c 相乘得到 ac。
• Find two numbers p and q such that p × q = ac and p + q = b.
• 找到两个数 p 和 q，使得 p × q = ac 且 p + q = b。
• Rewrite bx as px + qx and group pairs of terms.
• 将 bx 改写为 px + qx，并对项进行两两分组。
• Factor out the common factor from each group.
• 从每组中提出公因式。

For 2x² + 7x + 3, we rewrite it as 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1). Always expand your answer to check.

对于 2x² + 7x + 3，我们将其改写为 2x² + x + 6x + 3 =

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