Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in IGCSE Mathematics, from factorisation to the quadratic formula. This revision guide covers every method you need, with worked examples and exam tips to help you secure full marks.

二次方程在IGCSE数学中频繁出现，从因式分解到求根公式。本复习指南涵盖你需要的所有方法，配有例题和考试技巧，帮助你拿到满分。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0. The highest power of x is 2. Linear equations have x to the power 1, but quadratics include the squared term.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a ≠ 0。x 的最高次数是2。一次方程中 x 是一次幂，而二次方程包含平方项。

Examples of quadratic equations:

二次方程的例子：

x² − 5x + 6 = 0
2x² + 3x − 2 = 0
x² = 9 (which can be written as x² − 9 = 0)

x² − 5x + 6 = 0
2x² + 3x − 2 = 0
x² = 9（可写成 x² − 9 = 0）

You will solve quadratics using three main techniques: factorising, completing the square, and the quadratic formula.

你将使用三种主要技巧解二次方程：因式分解、配方法和求根公式。

2. Expanding and Factorising | 展开与因式分解

Before solving, you must be comfortable expanding double brackets: (x + p)(x + q) = x² + (p + q)x + pq. Conversely, factorising means writing a quadratic as a product of two brackets.

在解题之前，你必须熟练展开双括号：(x + p)(x + q) = x² + (p + q)x + pq。反过来，因式分解就是把二次式写成两个括号的乘积。

Example: Factorise x² + 7x + 12. Find two numbers that multiply to 12 and add to 7. These are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4).

例：因式分解 x² + 7x + 12。找两个数相乘得12，相加得7。这两个数是3和4，所以 x² + 7x + 12 = (x + 3)(x + 4)。

When the coefficient of x² is not 1, use the method of grouping or simply test factor pairs. For example, 2x² + 7x + 3 = (2x + 1)(x + 3).

当 x² 的系数不为1时，可使用分组法或直接尝试因数对。例如，2x² + 7x + 3 = (2x + 1)(x + 3)。

3. Solving by Factorising | 用因式分解解方程

If a quadratic can be factorised, solving is quick. The principle is that if AB = 0, then either A = 0 or B = 0.

如果二次式可以因式分解，求解就很快。原理是：若 AB = 0，则 A = 0 或 B = 0。

Example: Solve x² − 5x + 6 = 0.

例：解方程 x² − 5x + 6 = 0。

(x − 2)(x − 3) = 0

So x = 2 or x = 3

因此 x = 2 或 x = 3

Always rearrange the equation into the form ax² + bx + c = 0 before factorising. Do not divide both sides by x, because you may lose the solution x = 0.

因式分解前务必把方程整理成 ax² + bx + c = 0 的形式。不要两边同时除以 x，否则可能丢失解 x = 0。

4. Solving by Completing the Square | 用配方法解方程

Completing the square rewrites a quadratic as (x + p)² + q. This is useful for solving equations that do not factorise easily.

配方法将二次式改写为 (x + p)² + q。这适用于解不易因式分解的方程。

Example: Solve x² + 6x + 2 = 0 by completing the square.

例：用配方法解 x² + 6x + 2 = 0。

Take half of 6, which is 3, and write (x + 3)² = x² + 6x + 9. Then x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.

取6的一半，即3，写成 (x + 3)² = x² + 6x + 9。于是 x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。

(x + 3)² − 7 = 0

(x + 3)² = 7

x + 3 = ±√7

x = −3 ± √7

The exact solutions are x = −3 + √7 and x = −3 − √7.

精确解为 x = −3 + √7 和 x = −3 − √7。

5. Solving by the Quadratic Formula | 用求根公式解方程

The quadratic formula solves any quadratic equation ax² + bx + c = 0:

求根公式可以解任意二次方程 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / (2a)

You may be given this formula in the exam, but you must know how to substitute correctly.

考试中会提供此公式，但你必须知道如何正确代入。

Example: Solve 2x² + 3x − 5 = 0 using the formula. Here a = 2, b = 3, c = −5.

例：用公式解 2x² + 3x − 5 = 0。这里 a = 2，b = 3，c = −5。

x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2)

x = (−3 ± √(9 + 40)) / 4 = (−3 ± √49) / 4

x = (−3 + 7) / 4 = 1 or x = (−3 − 7) / 4 = −2.5

x = (−3 + 7) / 4 = 1 或 x = (−3 − 7) / 4 = −2.5

Always write the values of a, b, c first, then substitute carefully.

务必先写出 a、b、c 的值，再仔细代入。

6. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells us how many real roots a quadratic equation has.

表达式 b² − 4ac 称为判别式。它告诉我们二次方程有多少个实数根。

Discriminant	Number of real roots
b² − 4ac > 0	Two distinct real roots
b² − 4ac = 0	One repeated real root
b² − 4ac < 0	No real roots

判别式	实数根个数
b² − 4ac > 0	两个不同的实数根
b² − 4ac = 0	一个重根
b² − 4ac < 0	没有实数根

Example: For 3x² − 2x + 1 = 0, b² − 4ac = (−2)² − 4 × 3 × 1 = 4 − 12 = −8 < 0, so there are no real roots.

例：对于 3x² − 2x + 1 = 0，b² − 4ac = (−2)² − 4 × 3 × 1 = 4 − 12 = −8 < 0，因此没有实数根。

7. Graphical Interpretation | 图像解释

The solutions of a quadratic equation are the x-intercepts of the graph of y = ax² + bx + c. The graph is a parabola that opens upwards if a > 0 and downwards if a < 0.

二次方程的解就是 y = ax² + bx + c 的图像与 x 轴交点的横坐标。图像是抛物线，当 a > 0 时开口向上，当 a < 0 时开口向下。

If the discriminant is positive, the parabola crosses the x-axis at two points. If it is zero, the parabola touches the x-axis at one point. If it is negative, the parabola does not meet the x-axis.

如果判别式为正，抛物线与 x 轴相交于两点。如果为零，抛物线在一点处与 x 轴相切。如果为负，抛物线不与 x 轴相交。

Exam questions may ask you to sketch the graph. Remember to label the y-intercept (0, c) and the axis of symmetry x = −b/(2a).

考试题可能要求你画草图。记得标出 y 轴截距 (0, c) 和对称轴 x = −b/(2a)。

8. Solving Word Problems | 解应用题

Quadratic equations often arise from area, speed, or number problems. Set up the equation carefully, solve it, and check whether both solutions make sense in context.

二次方程常出现在面积、速度或数字问题中。仔细设方程，求解，并检查两个解在情境中是否都合理。

Example: A rectangle has length 4 cm more than its width. Its area is 45 cm². Find the width.

例：一个长方形的长比宽多4 cm，面积为45 cm²。求宽。

Let width = x. Then length = x + 4. So x(x + 4) = 45.

设宽为 x。则长为 x + 4。所以 x(x + 4) = 45。

x² + 4x − 45 = 0

(x + 9)(x − 5) = 0

x = −9 or x = 5

x = −9 或 x = 5

Width cannot be negative, so the width is 5 cm.

宽不能为负，所以宽为5 cm。

9. Common Mistakes | 常见错误

Here are common errors students make with quadratics:

以下是学生解二次方程时常犯的错误：

Forgetting to rearrange the equation to ax² + bx + c = 0 before factorising.
Dividing both sides by x and losing the root x = 0.
Wrong signs when substituting into the quadratic formula.
Mistaking (x + p)² for x² + p².

因式分解前忘记将方程整理成 ax² + bx + c = 0。
两边除以 x，丢失根 x = 0。
代入求根公式时符号出错。
把 (x + p)² 误认为 x² + p²。

Always check your answers by substituting them back into the original equation.

务必把解代回原方程检验。

10. Practice Questions | 练习题目

Try these questions yourself before checking the answers.

请先自己尝试以下题目，再核对答案。

1. Solve x² − 2x − 8 = 0 by factorising.

1. 用因式分解解 x² − 2x − 8 = 0。

2. Solve x² + 4x + 1 = 0 by completing the square. Leave your answer in surd form.

2. 用配方法解 x² + 4x + 1 = 0，答案保留根号形式。

3. Use the quadratic formula to solve 3x² − 5x − 2 = 0.

3. 用求根公式解 3x² − 5x − 2 = 0。

Answers:

答案：

1. x = 4 or x = −2

1. x = 4 或 x = −2

2. x = −2 ± √3

3. x = 2 or x = −1/3

3. x = 2 或 x = −1/3

11. Exam Tips | 考试技巧

In the Edexcel IGCSE exam, show all your working. If you use the quadratic formula, write down the values of a, b and c first. If you factorise, show the bracket step explicitly.

在爱德思IGCSE考试中，要写出全部过程。如果使用求根公式，先写出 a、b、c 的值。如果因式分解，要明确写出括号步骤。

When a question says “give your answers correct to two decimal places”, use the quadratic formula and round at the end. For exact answers, leave surds in simplified form.

当题目要求”答案保留两位小数”时，使用求根公式并在最后四舍五入。要求精确答案时，根号要化为最简形式。

Finally, always check whether your solutions fit the context, especially in word problems.

最后，务必检查解是否符合题意，尤其是在应用题中。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations | 解二次方程