Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

Quadratic equations are one of the most important topics in the Edexcel IGCSE Mathematics syllabus. They appear in algebra, graphs, geometry, and problem solving, and they form the foundation for many higher-level topics. Mastering factorisation, completing the square, and the quadratic formula is essential for achieving a high grade.

二次方程是 Edexcel IGCSE 数学大纲中最重要的主题之一。它们出现在代数、图像、几何和问题求解中，并为许多更高阶的内容奠定基础。熟练掌握因式分解、配方法和求根公式，是取得高分的关键。

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

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。

The coefficient a cannot be zero, because the term ax² is what makes the equation “quadratic”. If a = 0, the equation reduces to a linear equation, which has only one solution instead of two.

系数 a 不能为零，因为 ax² 项正是方程具有”二次”性质的原因。若 a = 0，方程就退化为一次方程，只会有一个解而不是两个。

A solution of a quadratic equation is called a root. A quadratic equation can have two distinct roots, one repeated root, or no real roots at all, depending on the value of the discriminant.

二次方程的解称为根。根据判别式的取值，二次方程可以有两个不同的根、一个重根，或者根本没有实数根。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the expression ax² + bx + c can be written as a product of two linear factors. For the Edexcel IGCSE exam, this is often the expected method when the coefficients are integers.

当 ax² + bx + c 可以写成两个一次因式的乘积时，因式分解是最快捷的方法。在 Edexcel IGCSE 考试中，若系数为整数，这通常是期望使用的方法。

Example: solve x² − 5x + 6 = 0. Look for two numbers that multiply to give 6 and add to give −5. These numbers are −2 and −3, so x² − 5x + 6 = (x − 2)(x − 3).

例如：解 x² − 5x + 6 = 0。寻找两个数，相乘得 6，相加得 −5。这两个数是 −2 和 −3，因此 x² − 5x + 6 = (x − 2)(x − 3)。

(x − 2)(x − 3) = 0

By the zero-product property, if a product is zero then at least one factor must be zero, so x − 2 = 0 or x − 3 = 0. Hence x = 2 or x = 3. Always write both answers clearly.

根据零积性质，若乘积为零，则至少有一个因式为零，所以 x − 2 = 0 或 x − 3 = 0。因此 x = 2 或 x = 3。务必清楚写出两个解。

When the coefficient of x² is not 1, the factorisation is slightly longer but follows the same idea. For example, 2x² + 5x − 3 = (2x − 1)(x + 3), which gives x = ½ or x = −3.

当 x² 的系数不为 1 时，因式分解稍长一些，但思路相同。例如 2x² + 5x − 3 = (2x − 1)(x + 3)，由此得到 x = ½ 或 x = −3。

3. Solving by Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This form is useful for finding the vertex of a parabola and for solving equations that cannot be factorised easily.

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。这种形式有助于求抛物线的顶点，也能用来解不易因式分解的方程。

Example: solve x² + 6x + 2 = 0. Half of the coefficient of x is 3, and (x + 3)² = x² + 6x + 9. Since 9 is extra, we subtract it: x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.

例如：解 x² + 6x + 2 = 0。x 的系数的一半是 3，而 (x + 3)² = x² + 6x + 9。因为多出了 9，所以要减去：x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。

Setting (x + 3)² − 7 = 0 gives (x + 3)² = 7. Taking square roots of both sides gives x + 3 = ±√7, so x = −3 ± √7.

令 (x + 3)² − 7 = 0，得 (x + 3)² = 7。两边同时开平方，x + 3 = ±√7，所以 x = −3 ± √7。

x = −3 ± √7

This method always works, even when factorisation is impossible. In the non-calculator paper, it is often the cleanest way to give exact answers involving square roots.

即使无法因式分解，这种方法也始终有效。在不使用计算器的试卷中，它通常是给出含平方根精确答案的最清晰方式。

4. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation of the form ax² + bx + c = 0. It is the most reliable method and should be memorised exactly.

求根公式可以解任何形如 ax² + bx + c = 0 的二次方程。它是最可靠的方法，必须准确记忆。

x = (−b ± √(b² − 4ac)) / (2a)

Example: solve 2x² − 4x − 3 = 0. Here a = 2, b = −4 and c = −3. Substitute these values into the formula carefully.

例如：解 2x² − 4x − 3 = 0。此处 a = 2，b = −4，c = −3。将这些值仔细代入公式。

First compute the discriminant: b² − 4ac = (−4)² − 4 × 2 × (−3) = 16 + 24 = 40. Then x = (4 ± √40) / 4. Since √40 = 2√10, we simplify to x = 1 ± √10 / 2.

先计算判别式：b² − 4ac = (−4)² − 4 × 2 × (−3) = 16 + 24 = 40。于是 x = (4 ± √40) / 4。由于 √40 = 2√10，化简得 x = 1 ± √10 / 2。

A common trap is reading b as positive when the original term is negative. Always write a, b and c with their signs before substituting, and use brackets around negative values.

一个常见陷阱是当 b 项为负时仍把 b 当作正数。代入前务必连同符号写出 a、b、c，并在负值外加括号。

5. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells us the number and type of roots without fully solving the equation.

b² − 4ac 称为判别式。它无需完全解方程，就能告诉我们根的个数和类型。

If b² − 4ac > 0, the equation has two distinct real roots, and the graph of y = ax² + bx + c crosses the x-axis at two points.

若 b² − 4ac > 0，方程有两个不同的实数根，y = ax² + bx + c 的图像与 x 轴相交于两点。

If b² − 4ac = 0, the equation has exactly one repeated root, and the graph touches the x-axis at a single point.

若 b² − 4ac = 0，方程有一个重根，图像与 x 轴相切于一点。

If b² − 4ac < 0, the equation has no real roots, and the graph never meets the x-axis.

若 b² − 4ac < 0，方程没有实数根，图像不与 x 轴相交。

Discriminant b² − 4ac	Nature of roots	Graph meaning
Positive ( > 0 )	Two distinct real roots	Crosses the x-axis twice
Zero ( = 0 )	One repeated real root	Touches the x-axis once
Negative ( < 0 )	No real roots	Does not meet the x-axis

6. Word Problems | 应用题

Many IGCSE questions describe a real-life situation that leads to a quadratic equation. Start by defining the unknown with a letter, then translate the information into an equation.

许多 IGCSE 题目描述一个实际情境，最终会得到二次方程。先设未知数并赋予字母，再把题中信息转化为方程。

Example: a rectangle has length (x + 3) cm and width x cm. Its area is 28 cm². The area formula gives x(x + 3) = 28.

例如：一个长方形的长为 (x + 3) cm，宽为 x cm，面积为 28 cm²。由面积公式得 x(x + 3) = 28。

Expanding gives x² + 3x − 28 = 0. This factorises as (x + 7)(x − 4) = 0, so x = −7 or x = 4. Since a length cannot be negative, x = 4.

展开得 x² + 3x − 28 = 0，因式分解为 (x + 7)(x − 4) = 0，因此 x = −7 或 x = 4。由于长度不能为负，所以 x = 4。

Always check that your final answer makes sense in the original context. Reject any solution that gives a negative length, width, area or time, and state the correct one clearly.

务必检查最终答案是否符合原题情境。排除任何导致长度、宽度、面积或时间为负的解，并清楚写出正确的答案。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 are the x-coordinates of the points where the parabola crosses the x-axis.

y = ax² + bx +

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Solving Quadratic Equations | 二次方程求解