Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear in almost every IGCSE Mathematics paper, whether in non-calculator algebra, graph work, or problem-solving. Mastering the standard techniques, the quadratic formula, and the discriminant can turn this topic into a reliable source of marks.

一元二次方程几乎出现在每一份 IGCSE 数学试卷中，无论是非计算器的代数题、函数图像题，还是应用题。掌握标准解法、求根公式和判别式，能让你在这一考点上稳定得分。

1. What Is a Quadratic Equation? | 什么是一元二次方程

A quadratic equation is an equation in which the highest power of the unknown variable is 2. It contains an x² term and has no term with x raised to a higher power.

一元二次方程是未知数的最高次数为 2 的方程。它含有 x² 项，并且没有任何次数高于 2 的项。

The general form is written as:

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are constants. The condition a ≠ 0 is essential; if a = 0, the equation becomes linear, not quadratic.

其中 a、b、c 是常数。条件 a ≠ 0 至关重要；若 a = 0，方程就变成一次方程，而不是二次方程。

For example, 2x² − 3x + 5 = 0 is quadratic, but x² + x³ = 0 is not quadratic.

例如，2x² − 3x + 5 = 0 是二次方程，而 x² + x³ = 0 不是二次方程。

2. The Standard Form | 标准形式

Before solving any quadratic equation, you should rearrange it into the standard form ax² + bx + c = 0. This means all terms move to one side, with zero on the other.

在解任何一元二次方程之前，都应先将其整理成标准形式 ax² + bx + c = 0。也就是说，把所有项移到一边，让另一边等于 0。

Consider the equation:

3x² + 4 = 7x

Subtract 7x from both sides to obtain:

3x² − 7x + 4 = 0

Here a = 3, b = −7 and c = 4. Always check the signs of b and c carefully.

此时 a = 3，b = −7，c = 4。务必仔细检查 b 和 c 的正负号。

Sometimes the equation appears with fractions or brackets. For instance, x(x + 2) = 3 must first be expanded to x² + 2x = 3, then rewritten as x² + 2x − 3 = 0.

有时方程含有分数或括号。例如，x(x + 2) = 3 必须先展开为 x² + 2x = 3，再改写成 x² + 2x − 3 = 0。

3. Solving by Factorisation | 用因式分解求解

Factorisation is the quickest method when the quadratic has simple integer roots. You write the quadratic as a product of two linear factors, then set each factor to zero.

当二次方程具有简单的整数根时，因式分解是最快捷的方法。你把二次式写成两个一次因式的乘积，然后令每个因式为 0。

Example: Solve x² − 5x + 6 = 0.

例：解 x² − 5x + 6 = 0。

Look for two numbers whose product is 6 and whose sum is −5. The numbers are −2 and −3.

找两个数，使其乘积为 6、和为 −5。这两个数是 −2 和 −3。

(x − 2)(x − 3) = 0

Now set each factor to zero:

现在令每个因式为 0：

x − 2 = 0, so x = 2
x − 3 = 0, so x = 3

The solutions are x = 2 and x = 3. Always expand your factors to check your work.

解为 x = 2 和 x = 3。务必展开因式来检验结果。

When the coefficient of x² is not 1, you must pay extra attention. To solve 2x² + 5x + 2 = 0, look for factors in the form (2x + 1)(x + 2) = 0, giving x = −0.5 and x = −2.

当 x² 的系数不为 1 时，需要格外小心。要解 2x² + 5x + 2 = 0，可寻找形如 (2x + 1)(x + 2) = 0 的因式，得到 x = −0.5 和 x = −2。

4. Difference of Two Squares | 平方差公式

A special factorisation pattern appears when the quadratic has the form x² − k². This is called the difference of two squares.

当二次式为 x² − k² 的形式时，会出现一种特殊的因式分解模式，称为平方差公式。

x² − k² = (x − k)(x + k)

For example, x² − 25 = 0 can be solved by recognising 25 = 5².

例如，x² − 25 = 0 可以通过识别 25 = 5² 来求解。

(x − 5)(x + 5) = 0

Therefore x = 5 or x = −5. This pattern also works when k is a fraction or a square root, such as x² − 2 = 0, giving x = ±√2.

因此 x = 5 或 x = −5。当 k 是分数或根式时该公式同样适用，例如 x² − 2 = 0，得到 x = ±√2。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This is useful when factorisation is not possible and when you need to find turning points of graphs.

配方法将二次式改写成 (x + p)² + q 的形式。当因式分解不可行，或需要求图像顶点时，这种方法非常有用。

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

例：用配方法解 x² + 6x + 2 = 0。

Start by halving the coefficient of x. Since the coefficient is 6, half is 3.

首先取 x 系数的一半。因为系数是 6，一半是 3。

(x + 3)² − 9 + 2 = 0

(x + 3)² − 7 = 0

Now isolate the square:

现在分离出完全平方项：

(x + 3)² = 7

Take the square root of both sides:

两边同时开平方：

x + 3 = ±√7

x = −3 ± √7

So the exact answers are x = −3 + √7 and x = −3 − √7. Leave roots in surd form unless the question asks for a decimal.

因此准确解为 x = −3 + √7 和 x = −3 − √7。除非题目要求小数，否则保留根式形式。

6. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation, even one that cannot be factorised. It is derived from completing the square and is given by:

求根公式可以解任意一元二次方程，包括无法因式分解的方程。它由配方法推导而来，公式为：

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, identify a, b and c from the standard form ax² + bx + c = 0, then substitute carefully.

使用公式时，先从标准形式 ax² + bx + c = 0 中确定 a、b、c，然后小心代入。

Example: Solve 2x² − 4x − 3 = 0.

例：解 2x² − 4x − 3 = 0。

Here a = 2, b = −4 and c = −3. Substitute into the formula:

这里 a = 2，b = −4，c = −3。代入公式：

x = (4 ± √(16 + 24)) / 4

x = (4 ± √40) / 4

x = (4 ± 2√10) / 4

x = 1 ± (√10) / 2

Write answers in the simplest surd form, then evaluate as decimals if required.

先写出最简根式形式，如需小数再进一步求值。

7. The Discriminant | 判别式

In the quadratic formula, the expression b² − 4ac is called the discriminant and is usually represented by Δ. It tells you how many real roots exist without solving the equation fully.

在求根公式中，b² − 4ac 称为判别式，通常用 Δ 表示。它能在不完全解方程的情况下，判断实数根的个数。

Value of Δ	Nature of roots	Δ 的值	根的情况
Δ > 0	Two distinct real roots	Δ > 0	两个不相等的实数根
Δ = 0	Two equal real roots	Δ = 0	两个相等实数根（重根）
Δ < 0	No real roots	Δ < 0	无实数根

For the equation x² − 6x + 9 = 0, the discriminant is (−6)² − 4(1)(9) = 36 − 36 = 0, so it has exactly one repeated root, x = 3.

对方程 x² − 6x + 9 = 0，判别式为 (−6)² − 4(1)(9) = 36 − 36 = 0，因此它只有一个重根 x = 3。

In past paper questions, you may be asked to find the value of k for which an equation has no real roots. Set Δ < 0 and solve the resulting inequality.

在历年真题中，你可能会被要求求使方程无实数根时 k 的值。此时令 Δ < 0，再解对应不等式。

8. Solving Word Problems | 解应用题

Quadratic equations frequently appear in area, product, and number problems. The key is to translate the words into algebra carefully, then solve and check that the answers make sense in the context.

二次方程经常出现在面积、乘积和数字问题中。关键是将文字准确地转化为代数式，然后求解，并检验答案是否符合题意。

Example: The area of a rectangle is 24 cm² and its length is 5 cm more than its width. Find the width.

例：一个长方形的面积为 24 平方厘米，长比宽多 5 厘米。求宽。

Let the width be w. Then the length is w + 5, so:

设宽为 w，则长为 w + 5，因此：

w(w + 5) = 24

w² + 5w − 24 = 0

Factorise: (w + 8)(w − 3) = 0, so w = −8 or w = 3. Since a length cannot be negative, the width is 3 cm and the length is 8 cm.

因式分解得 (w + 8)(w − 3) = 0，所以 w = −8 或 w = 3。由于长度不能为负，宽为 3 厘米，长为 8 厘米。

Always reject negative or impossible roots in geometry questions, but keep them in pure algebra problems when asked for all solutions.

在几何题目中一定要舍弃负数或不可能的根，但在纯代数题中，如果要求所有解，则要保留。

9. Graphing Quadratics | 二次函数图像

The graph of y = ax² + bx + c is a parabola. When a > 0, the curve opens upwards; when a < 0, it opens downwards.

y = ax² + bx + c 的图像是抛物线。当 a > 0 时，开口向上；当 a < 0 时，开口向下。

The roots of the equation ax² + bx + c = 0 are the x-intercepts of the graph. If the discriminant is negative, the graph does not cut the x-axis at all.

方程 ax² + bx + c = 0 的根就是图像与 x 轴的交点横坐标。如果判别式为负，图像与 x 轴没有交点。

The turning point can be found from the completed square form (x + p)² + q. For y = (x + p)² + q, the minimum point is at (−p, q). This is an essential skill for IGCSE graph questions.

顶点可以通过配方法得到的 (x + p)² + q 形式找到。对于 y = (x + p)² + q，最小值点为 (−p, q)。这是 IGCSE 图像题的必备技能。

For example, y = (x − 2)² − 4 has a minimum at (2, −4) and crosses the x-axis at x = 0 and x = 4.

例如，y = (x − 2)² − 4 的顶点在 (2, −4)，与 x 轴的交点为 x = 0 和 x = 4。

10. Common Mistakes and Tips | 常见错误与技巧

Many students lose marks because of careless sign errors. When substituting b into the quadratic formula, remember that if b = −6, then −b = 6 and b² = 36. Take your time with each substitution.

许多学生因为粗心的符号错误而失分。将 b 代入求根公式时，如果 b = −6，则 −b = 6，b² = 36。每一步代入都要放慢速度。

Another common mistake is forgetting to rearrange the equation into standard form before factorising. For example, x² = 7x should become x² − 7x = 0, not x² = 7x, before solving.

另一个常见错误是忘记在因式分解前先将方程整理成标准形式。例如，x² = 7x 应先变成 x² − 7x = 0，再求解。

Always check whether a = 0; if so, the equation is linear.
Reject impossible negative roots in real-world contexts.
Give exact surd answers unless decimals are requested.
Verify your roots by substituting back into the original equation.

始终检查是否 a = 0；如果是，方程为一次方程。
在实际问题情境中舍弃不可能的负根。
除非题目要求小数，否则给出精确根式答案。
将根代回原方程来检验答案。

Finally, practise with past papers under timed conditions. Factorisation, completing the square, and the quadratic formula are tools; using them fluently under pressure is the real exam skill.

最后，在限时条件下多做历年真题。因式分解、配方和求根公式都是工具；在考试压力下熟练使用它们才是真正的应试技能。

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Solving Quadratic Equations | 解一元二次方程