Quadratic Equations: Solving, Graphing and Applications | 二次方程：求解、图像与应用

📚 Quadratic Equations: Solving, Graphing and Applications | 二次方程：求解、图像与应用

A quadratic equation is a polynomial equation of degree two. It appears throughout IGCSE Mathematics, from algebraic manipulation to coordinate geometry and real-world modelling. Understanding how to solve quadratics by factorising, completing the square, and using the quadratic formula gives you a flexible toolkit for both routine and unfamiliar problems.

二次方程是一个二次多项式方程。它在 IGCSE 数学中无处不在，从代数运算到坐标几何和实际建模都会涉及。掌握因式分解法、配方法和求根公式三种求解方法，能让你在面对常规题和陌生题时都拥有灵活的工具箱。

1. Definition and Standard Form | 定义与标准形式

A quadratic equation in one variable is any equation that can be expressed as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is essential; if a were zero, the highest power of x would be 1, making the equation linear rather than quadratic. This standard form is the starting point for factorising, completing the square and using the quadratic formula.

一元二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。a ≠ 0 这一条件至关重要；如果 a 等于零，x 的最高次数就会变成 1，方程也就成为线性方程。这种标准形式是因式分解法、配方法和求根公式的共同起点。

ax² + bx + c = 0, where a ≠ 0

ax² + bx + c = 0，其中 a ≠ 0

Missing terms are common in IGCSE questions. For example, x² − 16 = 0 has b = 0, while 3x² + 7x = 0 has c = 0. Both are still quadratic because the x² term is present. Recognising missing terms helps you choose a more efficient method, such as difference of two squares or simple common-factor extraction.

缺失项在 IGCSE 考题中很常见。例如，x² − 16 = 0 中 b = 0，而 3x² + 7x = 0 中 c = 0。两者仍然是二次方程，因为 x² 项存在。能识别缺失项有助于你选择更高效的方法，例如平方差公式或简单的公因式提取。

2. Solving by Factorising | 因式分解法

Factorising is often the fastest method when a quadratic has simple integer factors. The method depends on the zero-product property: if pq = 0, then p = 0 or q = 0. After factorising the quadratic into two linear brackets, set each bracket equal to zero and solve for x.

当二次方程具有简单的整数因子时，因式分解法通常是最快的方法。该方法依赖于零乘积性质：如果 pq = 0，那么 p = 0 或 q = 0。将二次方程分解为两个一次括号后，令每个括号等于零并解出 x。

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 或 x = 3

Always rearrange the equation so that one side is zero before factorising. For example, x² + 2x = 8 must first become x² + 2x − 8 = 0, then (x + 4)(x − 2) = 0. Do not factorise the non-zero side alone, because the zero-product property only works when the product equals zero.

在因式分解之前，一定要先把方程整理为一边等于零的形式。例如，x² + 2x = 8 必须首先变为 x² + 2x − 8 = 0，然后得到 (x + 4)(x − 2) = 0。不要只对非零的一边进行因式分解，因为零乘积性质只在乘积等于零时才成立。

3. Solving by Completing the Square | 配方法

Completing the square is a powerful algebraic technique that rewrites x² + bx in the form (x + p)² − q. The method is especially useful when the roots involve surds or when you are asked for the turning point of a quadratic graph. It also leads directly to the quadratic formula.

配方法是一种强大的代数技巧，可以将 x² + bx 改写为 (x + p)² − q 的形式。当根包含无理根式或需要求二次图像顶点时，这种方法尤其有用。它还可以直接推导出求根公式。

x² + 8x + 3 = 0 → (x + 4)² − 16 + 3 = 0 → (x + 4)² = 13 → x = −4 ± √13

x² + 8x + 3 = 0 → (x + 4)² − 16 + 3 = 0 → (x + 4)² = 13 → x = −4 ± √13

If a ≠ 1, factor out a from the x² and x terms first, then complete the square inside the bracket. After that, expand carefully when finding the turning point. This

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