Quadratic Equations | 二次方程

📚 Quadratic Equations | 二次方程

Quadratic equations are a fundamental topic in IGCSE Mathematics. They appear in algebra, graph work, and problem-solving. This article explains the core methods for solving quadratic equations, with clear worked examples and key exam tips.

二次方程是IGCSE数学的基础内容，贯穿代数、函数图像和实际应用题。本文将系统讲解解二次方程的核心方法，并配以清晰的例题和考试要点。

1. General Form | 一般形式

A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0. The values of a, b, and c are real numbers, and x is the unknown. If a = 0, the equation becomes linear, not quadratic.

二次方程是指形如 ax² + bx + c = 0 的方程，其中 a ≠ 0。a、b、c 为实数，x 为未知数。若 a = 0，则方程退化为一次方程，不再是二次方程。

The three standard solution methods are: factorising, using the quadratic formula, and completing the square. You should know when each method is most efficient.

三种标准解法是：因式分解法、公式法、配方法。你需要知道每种方法在什么情况下最有效。

2. Solving by Factorising | 因式分解法

If the quadratic expression can be written as a product of two linear factors, then we can solve by setting each factor equal to zero. This method works best when the roots are rational numbers.

如果二次式能写成两个一次因式的乘积，则可令每个因式等于零来求解。当根为有理数时，此方法最快捷。

Example: Solve x² − 5x + 6 = 0.

示例：解方程 x² − 5x + 6 = 0。

(x − 2)(x − 3) = 0

So x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

因此 x − 2 = 0 或 x − 3 = 0，解得 x = 2 或 x = 3。

Always expand your factors to check: (x − 2)(x − 3) = x² − 3x − 2x + 6 = x² − 5x + 6.

务必展开因式进行验证：(x − 2)(x − 3) = x² − 3x − 2x + 6 = x² − 5x + 6。

3. Special Cases | 特殊情形

Some quadratics are differences of two squares, such as x² − 9 = (x − 3)(x + 3). Others are perfect squares, such as x² + 6x + 9 = (x + 3)². Recognising these patterns saves time.

有些二次式是平方差，如 x² − 9 = (x − 3)(x + 3)；有些是完全平方式，如 x² + 6x + 9 = (x + 3)²。识别这些模式可以节省时间。

Difference of two squares: a² − b² = (a − b)(a + b)
Perfect square: a² ± 2ab + b² = (a ± b)²

平方差：a² − b² = (a − b)(a + b)
完全平方：a² ± 2ab + b² = (a ± b)²

Example: Solve 4x² − 25 = 0.

示例：解方程 4x² − 25 = 0。

(2x − 5)(2x + 5) = 0

Hence x = 5/2 or x = −5/2.

因此 x = 5/2 或 x = −5/2。

4. Quadratic Formula | 求根公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the formula:

对于任意二次方程 ax² + bx + c = 0，解由以下公式给出：

x = (−b ± √(b² − 4ac)) / 2a

This formula works for all quadratics, including those that do not factorise easily. You should memorise it and know how to substitute values carefully.

此公式适用于所有二次方程，包括不易因式分解的情形。你需要牢记公式并熟练代入数值。

Example: Solve 2x² + 3x − 5 = 0 using the formula.

示例：用公式法解 2x² + 3x − 5 = 0。

Here a = 2, b = 3, c = −5. Substitute:

这里 a = 2，b = 3，c = −5。代入公式：

x = (−3 ± √(3² − 4×2×(−5))) / (2×2) = (−3 ± √(9 + 40)) / 4 = (−3 ± √49) / 4

So x = (−3 + 7)/4 = 1, or x = (−3 − 7)/4 = −5/2.

因此 x = (−3 + 7)/4 = 1，或 x = (−3 − 7)/4 = −5/2。

5. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells us how many real roots the equation has without solving it fully.

表达式 b² − 4ac 称为判别式。它能在不完全求解的情况下判断方程有几个实数根。

Value of b² − 4ac	Nature of roots
Positive and a perfect square	Two distinct rational roots
Positive, not a perfect square	Two distinct irrational roots
Zero	One repeated root
Negative	No real roots

在IGCSE考试中，你通常只需要判断“两个不等实根”、“两个相等实根”或“无实根”。

In IGCSE exams you are usually asked to identify two distinct real roots, two equal real roots, or no real roots.

6. Completing the Square | 配方法

Completing the square rewrites the quadratic as a(x + p)² + q. This form is useful for solving equations and for finding the vertex of a parabola.

配方法将二次式改写为 a(x + p)² + q 的形式。这种形式既可用于求解方程，也可用于求抛物线的顶点。

Example: Solve x² + 6x + 1 = 0 by completing the square.

示例：用配方法解 x² + 6x + 1 = 0。

(x + 3)² − 9 + 1 = 0 ⟹ (x + 3)² = 8

Take square roots: x + 3 = ±√8, so x = −3 ± 2√2.

两边开平方：x + 3 = ±√8，因此 x = −3 ± 2√2。

For an expression like 2x² − 8x + 5, first factor out the 2: 2(x² − 4x) + 5, then complete the square inside.

对于形如 2x² − 8x + 5 的式子，先提取 2：2(x² − 4x) + 5，再在括号内配方。

7. Solving Quadratic Inequalities | 二次不等式

Quadratic inequalities are solved by first finding the roots and then testing intervals. The graph of the quadratic helps visualise the solution set.

解二次不等式时，先求出对应方程的根，再测试各区间。借助二次函数图像可以更直观地判断解集。

Example: Solve x² − 4 > 0.

示例：解不等式 x² − 4 > 0。

First factor: (x − 2)(x + 2) > 0. The roots are 2 and −2. Test x = 0: (0)² − 4 = −4 < 0, so the middle region is not in the solution. Therefore x < −2 or x > 2.

先因式分解：(x − 2)(x + 2) > 0。根为 2 和 −2。代入 x = 0 得 (0)² − 4 = −4 < 0，因此中间区间不属于解集。所以 x < −2 或 x > 2。

8. Simultaneous Equations with a Quadratic | 二次联立方程

Intersection problems often lead to a linear and a quadratic equation. Solve by substitution: rearrange the linear equation to express one variable, then substitute into the quadratic.

求交点问题时常常会得到一个一次方程和一个二次方程。用代入法求解：先由一次方程表示一个变量，再代入二次方程。

Example: Solve y = x + 2 and y = x².

示例：解方程组 y = x + 2 和 y = x²。

Substitute: x + 2 = x² ⟹ x² − x − 2 = 0 ⟹ (x − 2)(x + 1) = 0. So x = 2 or x = −1. Then y = 4 or y = 1.

代入得：x + 2 = x² ⟹ x² − x − 2 = 0 ⟹ (x − 2)(x + 1) = 0。因此 x = 2 或 x = −1。对应 y = 4 或 y = 1。

9. Word Problems | 应用题

Many exam problems are set in real-world contexts, such as area, projectile motion, or number puzzles. Always define the unknown, form the quadratic equation, and interpret the roots in context.

许多考试题目以真实情境为背景，如面积、抛体运动或数字谜题。务必先设定未知数，列出二次方程，再结合情境解释根的含义。

Example: The length of a rectangle is 3 cm longer than its width. The area is 40 cm². Find the width.

示例：一个长方形的长比宽长 3 cm，面积为 40 cm²。求宽。

Let the width be w cm. Then length = w + 3, so w(w + 3) = 40 ⟹ w² + 3w − 40 = 0 ⟹ (w + 8)(w − 5) = 0. Since w > 0, w = 5. Width = 5 cm.

设宽为 w cm，则长为 w + 3，所以 w(w + 3) = 40 ⟹ w² + 3w − 40 = 0 ⟹ (w + 8)(w − 5) = 0。因为 w > 0，取 w = 5。宽为 5 cm。

10. Graphing Quadratics | 二次函数图像

The graph of y = ax² + bx + c is a parabola. The sign of a tells you whether it opens upward (a > 0) or downward (a < 0). The roots are the x-intercepts, and the vertex lies on the axis of symmetry x = −b/(2a).

y = ax² + bx + c 的图像是抛物线。a 的符号决定开口方向：a > 0 开口向上，a < 0 开口向下。根就是图像与 x 轴的交点，顶点位于对称轴 x = −b/(2a) 上。

The minimum or maximum value of the quadratic is found at the vertex. Completing the square makes it easy to read the vertex.

二次函数的最小值或最大值出现在顶点处。配方后可以直接读出顶点坐标。

11. Common Mistakes | 常见错误

Forgetting to set the equation equal to zero before factorising.
Losing the negative sign when substituting into the quadratic formula.
Dividing both sides by x, which loses a root.

因式分解前忘记把方程化为等于零的形式。
代入求根公式时遗漏负号。
两边同时除以 x，导致丢失一个根。

To avoid these, always check your answers by substitution into the original equation.

为避免这些错误，请将答案代回原方程进行验证。

12. Exam Tips | 考试要点

Attempt all questions. Show clear working steps. Use the quadratic formula when factorising is not obvious. Write your final answers correct to the required accuracy, usually three significant figures unless stated otherwise.

尝试回答所有问题，写出清晰的解题步骤。当因式分解不明显时，使用求根公式。最终答案按题目要求保留精度，通常为三位有效数字，除非另有说明。

Practise past papers to build speed and confidence. The more you practise, the more you will recognise patterns.

多做真题以提高速度和信心。练习越多，就越容易识别常见题型和解题模式。

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Quadratic Equations | 二次方程