Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations form the backbone of IGCSE Mathematics. They connect algebra, graphs, and problem-solving, and appear in almost every exam paper. Understanding how to solve them fluently is not optional — it is essential.

二次方程是IGCSE数学的基石。它将代数、图像和问题解决紧密相连，几乎出现在每一份考卷中。熟练求解二次方程不是可选项，而是必备技能。

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

A quadratic equation is any equation that can be written in the standard form:

二次方程是任何可以写成标准形式的方程：

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are real numbers, and x is the unknown. The highest power of x is 2, which makes the equation “quadratic”. For example, 3x² − 5x + 2 = 0 is quadratic, while x + 4 = 0 is linear and x³ − 1 = 0 is cubic.

这里，a、b、c 是实数，x 是未知数。x 的最高次数是2，因此这个方程被称为”二次”。例如，3x² − 5x + 2 = 0 是二次方程，而 x + 4 = 0 是一次方程，x³ − 1 = 0 是三次方程。

To solve a quadratic equation means to find all values of x that make the equation true. These values are called the roots or solutions of the equation.

解二次方程就是找出所有使方程成立的 x 值。这些值被称为方程的根或解。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer roots. The idea is to write the quadratic expression as a product of two linear factors, then use the zero-product property.

当二次方程具有简单的整数根时，因式分解是最快的方法。其思路是将二次表达式写成两个一次因式的乘积，然后利用零乘积性质。

Step-by-step example: Solve x² − 5x + 6 = 0.

分步示例：解 x² − 5x + 6 = 0。

1. Find two numbers that multiply to give 6 and add to give −5. These are −2 and −3.

1. 找到两个数，乘积为6，和为−5。这两个数是−2和−3。

2. Factorise: x² − 5x + 6 = (x − 2)(x − 3).

2. 因式分解：x² − 5x + 6 = (x − 2)(x − 3)。

3. Set each factor equal to zero: x − 2 = 0 or x − 3 = 0.

3. 令每个因式等于零：x − 2 = 0 或 x − 3 = 0。

4. Therefore x = 2 or x = 3.

4. 因此 x = 2 或 x = 3。

Always check whether the coefficient of x² is not 1. For example, 2x² − 5x − 3 = 0 factorises as (2x + 1)(x − 3) = 0, giving x = −1⁄2 or x = 3.

始终注意 x² 的系数是否不为1。例如，2x² − 5x − 3 = 0 可分解为 (2x + 1)(x − 3) = 0，得到 x = −1⁄2 或 x = 3。

In IGCSE exams, factorisation is worth using whenever the discriminant b² − 4ac is a perfect square. If not, use another method.

在IGCSE考试中，当判别式 b² − 4ac 是完全平方数时，应优先使用因式分解。如果不是，请使用其他方法。

3. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic in the form a(x + p)² + q = 0. This method always works and is especially useful for solving equations that do not factorise easily.

配方法将二次方程改写为 a(x + p)² + q = 0 的形式。这种方法始终有效，尤其适用于不易因式分解的方程。

Example: Solve x² + 6x + 5 = 0 by completing the square.

示例：用配方法解 x² + 6x + 5 = 0。

1. Move the constant term to the right: x² + 6x = −5.

1. 将常数项移到右边：x² + 6x = −5。

2. Take half of the coefficient of x, which is 3, and square it to get 9.

2. 取 x 系数的一半，即3，平方得到9。

3. Add 9 to both sides: x² + 6x + 9 = 4.

3. 两边同时加9：x² + 6x + 9 = 4。

4. Rewrite the left side as a square: (x + 3)² = 4.

4. 将左边写成完全平方：(x + 3)² = 4。

5. Take square roots: x + 3 = ±2, so x = −1 or x = −5.

5. 开平方：x + 3 = ±2，所以 x = −1 或 x = −5。

For equations with a ≠ 1, divide the whole equation by a first. For example, 2x² + 8x − 10 = 0 becomes x² + 4x − 5 = 0, then (x + 2)² = 9, giving x = 1 or x = −5.

对于 a ≠ 1 的方程，先将整个方程除以 a。例如，2x² + 8x − 10 = 0 变为 x² + 4x − 5 = 0，然后 (x + 2)² = 9，得到 x = 1 或 x = −5。

4. The Quadratic Formula | 二次公式

The quadratic formula is the most reliable method and works for every quadratic equation, including those with irrational roots.

二次公式是最可靠的方法，适用于所有二次方程，包括具有无理根的方程。

x = (−b ± √(b² − 4ac)) / 2a

Given ax² + bx + c = 0 with a ≠ 0, substitute the values of a, b, and c into the formula. The symbol ± means you perform the calculation twice: once with a plus sign and once with a minus sign.

对于 ax² + bx + c = 0（a ≠ 0），将 a、b、c 的值代入公式。符号 ± 表示计算两次：一次用加号，一次用减号。

Example: Solve 2x² − 4x − 3 = 0 using the quadratic formula.

示例：用二次公式解 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substitute:

这里 a = 2，b = −4，c = −3。代入：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± √10/2

So the solutions are x ≈ 2.58 or x ≈ −0.58 (to 2 decimal places). In IGCSE exams, always give your answers to the accuracy requested — often 3 significant figures.

因此解为 x ≈ 2.58 或 x ≈ −0.58（保留两位小数）。在IGCSE考试中，务必按题目要求的精度作答——通常保留3位有效数字。

Be careful with signs. A very common error is writing −b as b, or forgetting that b² is always non-negative. Write the formula down and substitute step by step.

注意符号。一个常见错误是把 −b 写成 b，或者忘记 b² 总是非负的。写出公式并逐步代入。

5. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant, often denoted by Δ. It tells us how many real roots the quadratic equation has — without solving it fully.

根号内的表达式 b² − 4ac 称为判别式，常用 Δ 表示。它告诉我们二次方程有多少个实数根——无需完全求解。

Discriminant Δ = b² − 4ac	Number of real roots	实数根的个数
Δ > 0	Two distinct real roots	两个不同的实数根
Δ = 0	One repeated real root	一个重根
Δ < 0	No real roots	没有实数根

Example: For x² + 4x + 5 = 0, Δ = 16 − 20 = −4. Since Δ < 0, this equation has no real roots; the graph never crosses the x-axis.

示例：对于 x² + 4x + 5 = 0，Δ = 16 − 20 = −4。因为 Δ < 0，该方程没有实数根；其图像不与 x 轴相交。

Example: For 9x² − 12x + 4 = 0, Δ = 144 − 144 = 0. There is exactly one root: x = 2⁄3.

示例：对于 9x² − 12x + 4 = 0，Δ = 144 − 144 = 0。方程恰有一个根：x = 2⁄3。

The discriminant is also used in problems where a condition is given, such as “the equation has two distinct roots, find the range of k“. Write the discriminant inequality and solve it.

判别式也用于给定条件的问题，例如”方程有两个不同的根，求 k 的取值范围”。写出判别式不等式并求解。

6. Graphs of Quadratic Functions | 二次函数图像

A quadratic function y = ax² + bx + c produces a curve called a parabola. When a > 0, the parabola opens upward (a “smile”); when a < 0, it opens downward (a "frown").

二次函数 y = ax² + bx + c 的图像称为抛物线。当 a > 0 时，抛物线开口向上；当 a < 0 时，开口向下。

Key features of the graph include:

图像的关键特征包括：

Roots: the x-coordinates where the parabola crosses the x-axis; these are the solutions to ax² + bx + c = 0.

根：抛物线与 x 轴交点的横坐标；它们是 ax² + bx + c = 0 的解。
y-intercept: the point (0, c), where the curve crosses the y-axis.

y 截距：点 (0, c)，即曲线与 y 轴的交点。
Vertex: the turning point of the parabola. Its x-coordinate is x = −b/(2a).

顶点：抛物线的转折点。其横坐标为 x = −b/(2a)。
Axis of symmetry: the vertical line x = −b/(2a), which divides the parabola into two mirror images.

对称轴：垂直线 x = −b/(2a)，将抛物线分成两个对称部分。

Example: For y = x² − 4x + 3, the axis of symmetry is x = 2. Substituting x = 2 gives y = −1, so the vertex is (2, −1). The roots are x = 1 and x = 3, and the y-intercept is (0, 3).

示例：对于 y = x² − 4x + 3，对称轴为 x = 2。将 x = 2 代入得 y = −1，所以顶点为 (2, −1)。根为 x = 1 和 x = 3，y 截距为 (0, 3)。

When sketching, plot the roots, the y-intercept, and the vertex first. Then draw a smooth symmetric curve through these points.

画草图时，先标出根、y 截距和顶点，然后通过这些点画一条光滑的对称曲线。

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

Many IGCSE questions present a real-world context that leads to a quadratic equation. The strategy is always the same: define a variable, form an equation from the given information, solve it, and check whether the answer makes sense in context.

许多IGCSE题目给出一个现实情境，最终可归结为二次方程。解题策略始终相同：设未知数，根据给定信息列出方程，求解，并检查答案在情境中是否合理。

Example: A rectangle has length 5 cm longer than its width. Its area is 84 cm². Find the width.

示例：一个矩形的长比宽长5 cm，面积为84 cm²。求宽。

Let the width be x cm. Then the length is (x + 5) cm, and:

设宽为 x cm，则长为 (x + 5) cm，于是：

x(x + 5) = 84 → x² + 5x − 84 = 0

Factorising gives (x + 12)(x − 7) = 0, so x = −12 or x = 7. Since a width cannot be negative, the answer is x = 7 cm.

因式分解得 (x + 12)(x − 7) = 0，所以 x = −12 或 x = 7。由于宽度不能为负，答案为 x = 7 cm。

Always reject negative solutions when the variable represents a length, time, or quantity. This interpretation step is often tested explicitly by examiners.

当未知数表示长度、时间或数量时，务必舍弃负数解。这一步的合理解释往往是考官明确考查的内容。

8. Common Mistakes and Exam Tips | 常见错误与考试技巧

Even strong students lose marks on quadratics due to small, avoidable errors. Here are the most frequent ones.

即使是优秀的学生也常因一些细微但可避免的错误在二次方程上失分。以下是最常见的几类。

Forgetting to rearrange to standard form: always write the equation as ax² + bx + c = 0 before applying factorisation or the formula.

忘记整理成标准形式：在因式分解或使用公式前，务必把方程写成 ax² + bx + c = 0。
Sign errors when substituting into the formula: use brackets when substituting negative values of b and c.

代入公式时出现符号错误：代入负的 b 和 c 时使用括号。
Dividing an equation by a variable: never divide both sides by x, because x = 0 may be a solution and you will lose it.

用变量去除方程：切勿两边同除以 x，因为 x = 0 可能是解，这样做会漏根。
Giving only one root: a quadratic equation always has two roots (counting repeats), unless the discriminant is negative.

只给一个根：二次方程通常有两个根（计重根），除非判别式为负。

Exam tips:

考试技巧：

Read the question carefully: if it says “answers to 3 significant figures”, use the quadratic formula rather than attempting to factorise.

仔细读题：如果题目要求”答案保留3位有效数字”，应使用二次公式而不是尝试因式分解。
Show your working clearly. Many IGCSE questions award method marks even if the final answer is wrong.

清晰展示解题过程。许多IGCSE题目即使最终答案错误，也会给方法分。
Check your solutions by substituting them back into the original equation. This takes 10 seconds and catches most mistakes.

将解代回原方程验证。这只需10秒，但能发现大部分错误。

Mastering quadratic equations opens the door to harder topics such as inequalities, functions, and the discriminant-based condition problems in later papers. Practise all three methods — factorisation, completing the square, and the quadratic formula — until each becomes automatic.

掌握二次方程为后续学习更难的话题打开了大门，如不等式、函数，以及后续试卷中基于判别式的条件问题。勤加练习三种方法——因式分解、配方和二次公式——直到每种方法都运用自如。

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Solving Quadratic Equations | 解二次方程