Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most important topics in the Edexcel IGCSE Mathematics syllabus. It appears in algebra, graphs, geometry, and problem-solving questions. Understanding how to solve quadratic equations fluently will help you gain confidence across many exam questions.

二次方程是 Edexcel IGCSE 数学考纲中最重要的主题之一。它出现在代数、图像、几何和应用题中。熟练掌握二次方程的解法，能帮助你在许多考试题目中建立信心。

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

A quadratic equation is an equation in which the highest power of the unknown variable is 2. The general form is written as ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.

二次方程是未知数最高次数为 2 的方程。它的一般形式写作 ax² + bx + c = 0，其中 a、b、c 是实数，且 a ≠ 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。

For example, 2x² + 5x – 3 = 0 is quadratic because the highest power of x is 2. In contrast, x + 3 = 0 is linear because the highest power is 1.

例如，2x² + 5x – 3 = 0 是二次方程，因为 x 的最高次数是 2。相比之下，x + 3 = 0 是一次方程，因为最高次数是 1。

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

Before solving a quadratic equation, it is often necessary to rearrange it into standard form ax² + bx + c = 0. This means moving all terms to one side of the equals sign and simplifying.

在解二次方程之前，往往需要将其整理成标准形式 ax² + bx + c = 0，也就是把所有项移到等号的一边并化简。

Roots are the solutions of the equation, i.e. the values of x that make the equation true.
系数是方程中的 a、b、c，它们可以是正数、负数或零，但 a 不能为零。
Solutions are also called the roots of the equation.

根是方程的解，也就是使方程成立的 x 的值。
系数是方程中的 a、b、c，它们可以是正数、负数或零，但 a 不能为零。
解也叫做方程的根。

For example, in the equation 3x² – 6x + 1 = 0, we have a = 3, b = -6 and c = 1. In the equation x² – 4 = 0, we have a = 1, b = 0 and c = -4.

例如，在方程 3x² – 6x + 1 = 0 中，a = 3，b = -6，c = 1。在方程 x² – 4 = 0 中，a = 1，b = 0，c = -4。

3. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic can be written as a product of two linear brackets. The principle is that if the product of two expressions is zero, then at least one of them must be zero.

因式分解法往往是当二次式可以写成两个线性括号相乘时最快的方法。其原理是：如果两个表达式的乘积为零，那么其中至少一个必须为零。

If AB = 0, then A = 0 or B = 0.

Example: Solve x² – 7x + 10 = 0.

示例：解 x² – 7x + 10 = 0。

Step 1: Factorise the left-hand side. We need two numbers that multiply to +10 and add to -7. These numbers are -2 and -5.

第一步：对左边进行因式分解。我们需要两个数，它们的积为 +10，和为 -7。这两个数是 -2 和 -5。

So x² – 7x + 10 = (x – 2)(x – 5) = 0.

因此 x² – 7x + 10 = (x – 2)(x – 5) = 0。

Then set each bracket equal to zero: x – 2 = 0 or x – 5 = 0, giving x = 2 or x = 5.

然后令每个括号等于零：x – 2 = 0 或 x – 5 = 0，得到 x = 2 或 x = 5。

Always check if the coefficient a is not 1. For example, solve 2x² + 5x – 3 = 0 by factorising.

当 a 不等于 1 时也要会处理。例如，用因式分解法解 2x² + 5x – 3 = 0。

We look for two numbers that multiply to 2 × (-3) = -6 and add to 5. These numbers are 6 and -1. Rewrite the middle term:

我们寻找两个数，它们的积为 2 × (-3) = -6，和为 5。这两个数是 6 和 -1。改写中间项：

2x² + 6x – x – 3 = 0, then factor by grouping:

2x² + 6x – x – 3 = 0，然后分组因式分解：

2x(x + 3) – 1(x + 3) = 0, so (2x – 1)(x + 3) = 0.

2x(x + 3) – 1(x + 3) = 0，所以 (2x – 1)(x + 3) = 0。

Therefore x = 1/2 or x = -3.

因此 x = 1/2 或 x = -3。

4. Solving by Completing the Square | 配方法

Completing the square is a powerful method that works for any quadratic, especially when factorisation is difficult. It rewrites the equation in the form (x + p)² = q.

配方法是一种适用于任何二次方程的强大方法，特别是在因式分解困难时尤其有用。它将方程改写成 (x + p)² = q 的形式。

The process starts with x² + bx. Add and subtract (b/2)² to create a perfect square.

这个过程从 x² + bx 开始。加上并减去 (b/2)² 以形成一个完全平方。

Example: Solve x² + 6x + 4 = 0 by completing the square.

示例：用配方法解 x² + 6x + 4 = 0。

First, move the constant to the other side: x² + 6x = -4.

首先，把常数项移到另一边：x² + 6x = -4。

Half of 6 is 3, so add 3² = 9 to both sides:

6 的一半是 3，所以两边加上 3² = 9：

x² + 6x + 9 = -4 + 9

Now the left side is a perfect square: (x + 3)² = 5.

现在左边是一个完全平方：(x + 3)² = 5。

Take square roots: x + 3 = ±√5.

开平方：x + 3 = ±√5。

So x = -3 ± √5, which gives two solutions: x = -3 + √5 or x = -3 – √5.

所以 x = -3 ± √5，即两个解：x = -3 + √5 或 x = -3 – √5。

If the coefficient of x² is not 1, first divide the whole equation by that coefficient.

如果 x² 的系数不是 1，先将整个方程除以该系数。

5. The Quadratic Formula | 求根公式

The quadratic formula is a universal method for solving any quadratic equation. It is derived from completing the square and is extremely important for the IGCSE exam.

求根公式是解任何二次方程的通用方法。它由配方法推导而来，在 IGCSE 考试中极其重要。

x = (-b ± √(b² – 4ac)) / (2a)

For the equation ax² + bx + c = 0, substitute the values of a, b and c into the formula.

对于方程 ax² + bx + c = 0，将 a、b、c 的值代入公式即可。

Example: Solve 2x² – 3x – 2 = 0 using the quadratic formula.

示例：用求根公式解 2x² – 3x – 2 = 0。

Here a = 2, b = -3, c = -2.

这里 a = 2，b = -3，c = -2。

Substitute into the formula:

代入公式：

x = (3 ± √((-3)² – 4 × 2 × (-2))) / (2 × 2)

x = (3 ± √(9 + 16)) / 4 = (3 ± √25) / 4 = (3 ± 5) / 4

So x = (3 + 5)/4 = 8/4 = 2, or x = (3 – 5)/4 = -2/4 = -1/2.

所以 x = (3 + 5)/4 = 8/4 = 2，或 x = (3 – 5)/4 = -2/4 = -1/2。

The quadratic formula is especially useful when the equation does not factorise easily.

当方程不易因式分解时，求根公式特别有用。

6. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: b² – 4ac. It tells us the number and type of solutions without solving the whole equation.

判别式是求根公式中根号下的部分：b² – 4ac。它告诉我们方程的解的数量和类型，而无需完整解方程。

Discriminant value	Nature of roots
b² – 4ac > 0	Two distinct real roots
b² – 4ac = 0	One repeated real root (equal roots)
b² – 4ac < 0	No real roots (complex roots)

判别式的值决定了根的情况：大于 0 时有两个不同的实数根；等于 0 时有一个重根；小于 0 时没有实数根。

For example, for x² + 2x + 5 = 0, the discriminant is 2² – 4 × 1 × 5 = 4 – 20 = -16, which is less than zero, so there are no real roots.

例如，对于 x² + 2x + 5 = 0，判别式为 2² – 4 × 1 × 5 = 4 – 20 = -16，小于零，因此没有实数根。

The discriminant also tells us whether the graph of y = ax² + bx + c crosses the x-axis, touches it, or does not meet it at all.

判别式还告诉我们 y = ax² + bx + c 的图像是与 x 轴相交、相切还是完全不相交。

7. Solving Quadratic Equations by Graph | 图像法

The solutions of a quadratic equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola y = ax² + bx + c. By drawing the graph accurately, you can read off the roots.

二次方程 ax² + bx + c = 0 的解对应于抛物线 y = ax² + bx + c 与 x 轴的交点。通过精确画图，你可以读出根的值。

To draw a quadratic graph, you can create a table of values for x and y, plot the points, and join them with a smooth curve.

要画二次函数图像，你可以先制作 x 和 y 的数值表，描点，然后用平滑曲线连接。

Example: Solve x² – 2x – 3 = 0 graphically.

示例：用图像法解 x² – 2x – 3 = 0。

The graph of y = x² – 2x – 3 crosses the x-axis at x = -1 and x = 3. These are the roots of the equation.

y = x² – 2x – 3 的图像在 x = -1 和 x = 3 处与 x 轴相交。这两个值就是方程的根。

If the equation is not in standard form, rearrange it first. Graphical solutions may be approximate if the roots are not integers.

如果方程不是标准形式，先整理。当根不是整数时，图像解可能是近似值。

8. Word Problems with Quadratics | 二次方程应用题

Many real-life problems can be solved by forming a quadratic equation. In exams, you may be asked to set up the equation and then solve it.

许多实际问题可以通过建立二次方程来解决。在考试中，你可能会被要求设出方程并求解。

Example: A rectangle has length (x + 3) cm and width (x – 1) cm. Its area is 21 cm². Find the value of x.

示例：一个长方形的长为 (x + 3) cm，宽为 (x – 1) cm，面积为 21 cm²。求 x 的值。

The area gives the equation (x + 3)(x – 1) = 21.

面积得到方程 (x + 3)(x – 1) = 21。

Expand: x² + 3x – x – 3 = 21, so x² + 2x – 3 = 21.

展开：x² + 3x – x – 3 = 21，所以 x² + 2x – 3 = 21。

Rearrange: x² + 2x – 24 = 0.

整理：x² + 2x – 24 = 0。

Factorise: (x + 6)(x – 4) = 0, so x = -6 or x = 4.

因式分解：(x + 6)(x – 4) = 0，所以 x = -6 或 x = 4。

Since lengths must be positive, x = 4 is the only valid answer.

因为长度必须为正，所以唯一合理的答案是 x = 4。

Always check that your solutions make sense in the context of the problem.

始终检查你的解在问题情境中是否有意义。

9. Common Mistakes to Avoid | 常见错误避免

Many students lose marks on quadratic equations due to avoidable errors. Here are some of the most common pitfalls and how to avoid them.

许多学生在二次方程上失分是因为可以避免的错误。以下是一些最常见的陷阱以及如何避免它们。

Forgetting to rearrange the equation into the form ax² + bx + c = 0 before factorising or using the formula.
Omitting the ± sign when taking square roots.
Misidentifying the values of a, b and c when a or b is negative.
Dividing both sides of an equation by a variable that could be zero, which may lose a solution.
Making sign errors when substituting into the quadratic formula.

在因式分解或使用公式前忘记将方程整理成 ax² + bx + c = 0 的形式。
开平方时遗漏 ± 符号。
当 a 或 b 为负数时，错误判断 a、b、c 的值。
将方程两边同时除以可能为零的变量，从而丢失一个解。
代入求根公式时出现符号错误。

Check your answers by substituting them back into the original equation.

通过将答案代回原方程来检验你的结果。

10. Practice Questions | 练习题

Try the following questions on your own, then check your answers. These are typical of Edexcel IGCSE style questions.

先自己尝试以下题目，再核对答案。这些是 Edexcel IGCSE 风格的典型题目。

Solve x² – 5x + 6 = 0.
Solve 3x² + 7x – 6 = 0.
Solve x² + 3x – 1 = 0 using the quadratic formula. Leave your answer in surd form.
Find the discriminant of 2x² – 4x + 3 and state the number of real roots.
The product of two consecutive odd integers is 143. Form a quadratic equation and solve it to find the integers.

解 x² – 5x + 6 = 0。
解 3x² + 7x – 6 = 0。
用求根公式解 x² + 3x – 1 = 0，答案保留根号形式。
求 2x² – 4x + 3 的判别式，并说明实数根的个数。
两个连续奇整数的积为 143。建立二次方程并求解，找出这两个整数。

Answers: 1) x = 2 or x = 3. 2) x = 2/3 or x = -3. 3) x = (-3 ± √13)/2. 4) Discriminant = -8, so no real roots. 5) The integers are 11 and 13, or -13 and -11.

答案：1) x = 2 或 x = 3。2) x = 2/3 或 x = -3。3) x = (-3 ± √13)/2。4) 判别式 = -8，所以没有实数根。5) 这两个整数是 11 和 13，或 -13 和 -11。

11. Summary | 总结

Quadratic equations are a core part of Edexcel IGCSE Mathematics. You should be confident in recognising the standard form, choosing an appropriate method, and solving accurately.

二次方程是 Edexcel IGCSE 数学的核心内容。你应该能够熟练识别标准形式，选择合适的方法，并准确求解。

Use factorisation when it is obvious, completing the square when needed, and the quadratic formula as a reliable fallback. Remember to check the discriminant to understand the nature of the roots.

当因式分解明显时使用因式分解法，需要时使用配方法，并将求根公式作为可靠的备用方法。记得检查判别式以了解根的性质。

With regular practice, quadratic equations will become one of the easiest topics on the exam.

通过定期练习，二次方程将变成考试中最容易的考点之一。

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Solving Quadratic Equations | 解二次方程