Quadratic Equations in IGCSE Mathematics: Methods, Graphs and Applications | IGCSE 数学二次方程：解法、图像与应用

📚 Quadratic Equations in IGCSE Mathematics: Methods, Graphs and Applications | IGCSE 数学二次方程：解法、图像与应用

Quadratic equations are one of the most heavily examined topics in IGCSE Mathematics. They link algebra, graphs, and real-world problem solving, so a strong command of quadratics can greatly improve your final grade. This article explains what quadratic equations are, how to solve them using three main methods, how to interpret their graphs, and how to avoid common exam errors.

二次方程是 IGCSE 数学中考查最频繁的主题之一。它们将代数、图像和实际问题联系在一起，因此扎实掌握二次方程能明显提高你的最终成绩。本文说明什么是二次方程、三种主要解法、如何理解图像以及如何避免考试中的常见错误。

---

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

tion becomes linear, not quadratic. The highest power of the unknown x is 2, which is why it is called quadratic.

二次方程是能写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。条件 a ≠ 0 至关重要，因为如果 a = 0，方程就变成一次方程而不是二次方程。未知数 x 的最高次数为 2，因此称为二次方程。

pers, quadratics may appear in different orders, with brackets that need expanding, or with fractions that need clearing before the standard form is visible.

例如，x² - 7x + 12 = 0 是二次方程，而 3x + 5 = 0 是一次方程。在 IGCSE 考试中，二次方程可能以不同顺序出现，可能带有需要先展开的括号，也可能含有需要先清除的分数，然后才能看出标准形式。

The word quadratic comes from the Latin word quadratus, meaning square. This makes sense because the variable x is squared in the equation.

二次方程这个词来源于拉丁语 quadratus，意思是平方。这很合理，因为方程中的变量 x 被平方了。

---

2. Standard Form and Key Terms | 标准形式与关键术语

ficient of x², b is the coefficient of x, and c is the constant term. Identifying these values correctly is the first step in many solution methods.

表达式 ax² + bx + c 称为二次式。在方程 ax² + bx + c = 0 中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。正确识别这些值是许多解法的第一步。

o the coefficient, so b is negative. In 4x² + 7x - 1 = 0, we have a = 4, b = 7 and c = -1.

例如，在 2x² - 5x + 3 = 0 中，a = 2，b = -5，c = 3。注意负号属于系数，因此 b 是负数。在 4x² + 7x - 1 = 0 中，a = 4，b = 7，c = -1。

The solutions of a quadratic equation are also called roots or zeros. A root is a value of x that makes the left-hand side equal to zero. A quadratic equation can have two real roots, one repeated real root, or no real roots.

二次方程的解也称为根或零点。根是使等式左边等于零的 x 值。二次方程可能有两个实根、一个重根或没有实根。

side is exactly 0. For example, x² - 3x = 10 should first be written as x² - 3x - 10 = 0.

在求解之前，一定要把所有项移到等式一边，使另一边恰好为 0。例如，x² - 3x = 10 应首先写成 x² - 3x - 10 = 0。

---

3. Solving by Factorisation | 因式分解法求解

Factorisation works when the quadratic expression can be written as a product of two linear factors. The key idea is the zero product property: if a product of two factors is zero, then at least one of the factors must be zero.

当二次式可以写成两个一次因式的乘积时，可以使用因式分解法。核心思想是零乘积性质：如果两个因式的乘积为零，那么至少有一个因式必须为零。

e (-2) × (-3) = 6 and (-2) + (-3) = -5. Therefore the equation becomes (x - 2)(x - 3) = 0.

例子：解 x² - 5x + 6 = 0。寻找两个数，乘积为 6，和为 -5。数字 -2 和 -3 符合，因为 (-2) × (-3) = 6，且 (-2) + (-3) = -5。因此方程变为 (x - 2)(x - 3) = 0。

l to zero: x - 2 = 0 or x - 3 = 0. This gives x = 2 or x = 3. These are the two solutions.

接下来，令每个因式等于零：x - 2 = 0 或 x - 3 = 0。得到 x = 2 或 x = 3。这就是两个解。

5) + 1(2x - 5) = 0, which gives (2x - 5)(x + 1) = 0. The solutions are x = 5/2 and x = -1.

对于非首一二次式，例如 2x² - 3x - 5 = 0，需要找到两个数，乘积为 a × c = 2 × (-5) = -10，和为 b = -3。数字 -5 和 2 符合，因此拆分中间项：2x² - 5x + 2x - 5 = 0。然后分组分解：x(2x - 5) + 1(2x - 5) = 0，得到 (2x - 5)(x + 1) = 0。解为 x = 5/2 和 x = -1。

Always check your factorisation by expanding the brackets. A quick expansion can catch sign errors before you write your final answer.

一定要通过展开括号来检验因式分解。快速展开可以在写出最终答案前发现符号错误。

---

广告 / Sponsored

4. Solving by Completing the Square | 配方法求解

Completing the square rewrites ax² + bx + c into the form a(x + p)² + q, where p and q are constants. This method is especially useful when the quadratic does not factorise neatly and when you need to find the vertex of a graph.

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式，其中 p 和 q 为常数。当二次式不能整齐分解以及需要求图像顶点时，这种方法特别有用。

to get 16. Add and subtract 16: x² + 8x + 16 - 16 - 3 = 0. This becomes (x + 4)² - 19 = 0.

例子：解 x² + 8x - 3 = 0。取 8 的一半为 4，平方得 16。加减 16：x² + 8x + 16 - 16 - 3 = 0。化为 (x + 4)² - 19 = 0。

Now isolate the square: (x + 4)² = 19. Take the square root of both sides: x + 4 = ±√19. Therefore x = -4 ± √19. The exact solutions are x = -4 + √19 and x = -4 - √19.

现在分离平方项：(x + 4)² = 19。对两边开平方：x + 4 = ±√19。因此 x = -4 ± √19。精确解为 x = -4 + √19 和 x = -4 - √19。

If the coefficient of x² is not 1, factor it out first. For example, in 2x² + 12x + 5, write 2(x² + 6x) + 5, then complete the square inside the bracket. Do not forget to multiply the added constant by the leading coefficient when simplifying.

如果 x² 的系数不是 1，需要先把它提出来。例如，在 2x² + 12x + 5 中，写成 2(x² + 6x) + 5，然后在括号内完成配方。化简时不要忘记将加上的常数乘以首项系数。

) is the vertex of the parabola. This is a very useful skill for graph sketching questions.

配方法还能得到顶点式 y = a(x - h)² + k，其中 (h, k) 是抛物线的顶点。这是画图题中非常有用的技巧。

---

5. The Quadratic Formula | 二次公式法

und using the quadratic formula. This formula works for all quadratic equations, including those with irrational or non-integer roots.

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）