Mastering Quadratic Equations | 掌握一元二次方程

📚 Mastering Quadratic Equations | 掌握一元二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, coordinate geometry and problem solving. Understanding how to solve quadratic equations fluently will give you a strong foundation for higher-level study.

一元二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像、坐标几何和实际问题解决中。熟练掌握一元二次方程的解法，将为你更高层次的学习打下坚实基础。

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

A quadratic equation is an equation of degree 2, which means the highest power of the variable is 2. The general form is written as:

一元二次方程是次数为 2 的方程，即变量的最高次数是 2。它的一般形式写成：

ax² + bx + c = 0, where a ≠ 0

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

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

For example, x² − 4x + 3 = 0 is quadratic, but 2x + 5 = 0 is not. A quadratic equation can have two, one or even no real solutions, depending on the values of a, b and c.

例如，x² − 4x + 3 = 0 是二次方程，而 2x + 5 = 0 不是。二次方程可以有两个、一个甚至没有实数解，这取决于 a、b、c 的值。

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

Before solving any quadratic equation, always rearrange it into the standard form ax² + bx + c = 0. This makes every method easier to apply and prevents errors. If the equation is written as x² = 3x + 4, you must first rewrite it as x² − 3x − 4 = 0.

在解任何二次方程之前，务必先将其整理成标准形式 ax² + bx + c = 0。这能让每种解法更容易应用，并避免错误。如果方程写成 x² = 3x + 4，你必须先改写为 x² − 3x − 4 = 0。

Key terms you should know:

你应该掌握的关键术语：

• Coefficient | 系数: The numbers multiplying the variable. In 2x², the coefficient of x² is 2.
• Constant term | 常数项: The number without a variable, such as 5 in x² + 3x + 5.
• Root / Solution | 根 / 解: A value of x that makes the equation equal to zero.
• Degree | 次数: The highest power of the variable, which is 2 for quadratics.

Being comfortable with these terms will help you follow exam questions that ask for “the roots of the equation” or “the solutions of the equation”.

熟悉这些术语，能帮助你理解考试题目中的要求，例如”求方程的根”或”求方程的解”。

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3. Solving by Factorisation | 因式分解法

Factorisation is usually the fastest method when the quadratic can be factored easily. For a quadratic with a = 1, such as x² + bx + c, you look for two numbers that multiply to give c and add to give b.

当一个二次方程容易分解时，因式分解通常是最快的方法。对于 a = 1 的二次方程，例如 x² + bx + c，你需要找到两个数，它们相乘等于 c，相加等于 b。

Solve x² + 5x + 6 = 0. The two numbers are 2 and 3, because 2 × 3 = 6 and 2 + 3 = 5. Therefore:

解 x² + 5x + 6 = 0。这两个数是 2 和 3，因为 2 × 3 = 6 且 2 + 3 = 5。因此：

(x + 2)(x + 3) = 0

Then set each bracket equal to zero: x + 2 = 0 gives x = −2, and x + 3 = 0 gives x = −3. Always state both solutions clearly.

然后令每个括号等于零：x + 2 = 0 得 x = −2，x + 3 = 0 得 x = −3。务必清楚写出两个解。

If c is negative, one number must be positive and the other negative. For example, x² − 2x − 8 = 0 factors as (x − 4)(x + 2) = 0, giving x = 4 or x = −2.

如果 c 是负数，则一个数为正，另一个为负。例如，x² − 2x − 8 = 0 分解为 (x − 4)(x + 2) = 0，得 x = 4 或 x = −2。

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4. Factorising Quadratics with a ≠ 1 | 首项系数不为 1 的因式分解

When the coefficient of x² is not 1, you must factorise carefully. One reliable method is to split the middle term. For 2x² + 7x + 3 = 0, multiply a and c: 2 × 3 = 6. Find two numbers that multiply to 6 and add to 7, which are 1 and 6.

当 x² 的系数不是 1 时，你必须仔细分解。一种可靠的方法是拆中项。对于 2x² + 7x + 3 = 0，将 a 和 c 相乘：2 × 3 = 6。找到两个数相乘为 6 且相加为 7，即 1 和 6。

Rewrite the equation as 2x² + x + 6x + 3 = 0, then factor in pairs:

将方程改写为 2x² + x + 6x + 3 = 0，然后分组分解：

x(2x + 1) + 3(2x + 1) = 0

Now take out the common bracket: (x + 3)(2x + 1) = 0. Hence x = −3 or x = −½.

现在提取公因式：(x + 3)(2x + 1) = 0。因此 x = −3 或 x = −½。

Another special case is the difference of two squares. The identity x² − a² = (x − a)(x + a) can be used directly. For example, x² − 9 = 0 becomes (x − 3)(x + 3) = 0, so x = ±3.

另一个特殊情形是平方差公式。恒等式 x² − a² = (x − a)(x + a) 可以直接使用。例如，x² − 9 = 0 变成 (x − 3)(x + 3) = 0，所以 x = ±3。

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5. Solving by Completing the Square | 配方法

Completing the square is a powerful method that works for every quadratic. The idea is to rewrite the quadratic as a perfect square plus (or minus) a constant. For x² + 6x + 2 = 0, take half of 6 to get 3, and write:

配方法是一种对任何二次方程都适用的强大方法。其思想是将二次式改写为一个完全平方加上（或减去）一个常数。对于 x² + 6x + 2 = 0，取 6 的一半得 3，然后写成：

(x + 3)² − 9 + 2 = 0

Simplify: (x + 3)² − 7 = 0. Then (x + 3)² = 7, so x + 3 = ±√7, giving x = −3 ± √7.

化简：(x + 3)² − 7 = 0。然后 (x + 3)² = 7，所以 x + 3 = ±√7，得 x = −3 ± √7。

The general rule is: x² + bx + c = (x + b/2)² − (b/2)² + c. This method is especially useful when the quadratic does not factorise, or when you need to find the turning point of a graph.

一般规则是：x² + bx + c = (x + b/2)² − (b/2)² + c。当二次式无法因式分解，或者需要求图像的顶点时，这种方法尤其有用。

Remember that when a ≠ 1, factor out a first. For 2x² − 8x + 5 = 0, write 2(x² − 4x) + 5 = 0, then complete the square inside the bracket.

记住当 a ≠ 1 时，先提出 a。对于 2x² − 8x + 5 = 0，写成 2(x² − 4x) + 5 = 0，然后在括号内配方。

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6. The Quadratic Formula | 求根公式

The quadratic formula is a universal method. For any quadratic equation in the form ax² + bx + c = 0, the solutions are given by:

求根公式是一种通用的方法。对于任何形式为 ax² + bx + c = 0 的二次方程，其解为：

x = (−b ± √(b² − 4ac)) / 2a

Worked example: solve 2x² − 3x − 2 = 0. Here a = 2, b = −3 and c = −2. Substitute into the formula:

示例：解 2x² − 3x − 2 = 0。这里 a = 2，b = −3，c = −2。代入公式：

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

This simplifies to x = (3 ± √(9 + 16)) / 4 = (3 ± √25) / 4 = (3 ± 5) / 4. Therefore x = 2 or x = −½.

这化简为 x = (3 ± √(9 + 16)) / 4 = (3 ± √25) / 4 = (3 ± 5) / 4。因此 x = 2 或 x = −½。

Always check the sign of b. A common mistake is to forget the negative sign when substituting. Write out every substitution step clearly to avoid errors.

始终注意 b 的符号。常见错误是在代入时忘记负号。写出每一步代入过程，以尽量避免错误。

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7. The Discriminant Δ | 判别式 Δ

The part inside the square root, b² − 4ac, is called the discriminant. It tells you how many real roots the equation has without solving it. In IGCSE exams you may be asked to “find the value of the discriminant” or “show that the equation has two distinct roots”.

根号内的部分 b² − 4ac 称为判别式。它能在不解方程的情况下告诉你方程有多少个实数根。在 IGCSE 考试中，你可能会被要求”求判别式的值”或”证明方程有两个不相等的实根”。

The rules are:

规则如下：

Discriminant | 判别式	Number of real roots | 实数根个数
b² − 4ac > 0	Two distinct real roots | 两个不相等的实数根
b² − 4ac = 0	One repeated real root | 一个相等的实数根
b² − 4ac < 0	No real roots | 没有实数根

For example, the equation x² + 2x + 5 = 0 has discriminant Δ = 4 − 20 = −16. Since this is negative, there are no real solutions.

例如，方程 x² + 2x + 5 = 0 的判别式 Δ = 4 − 20 = −16。由于这是负数，因此没有实数解。

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

The graph of y = ax² + bx + c is always a parabola. If a > 0, the parabola opens upward like a “U”. If a < 0, it opens downward like an upside-down "U".

y = ax² + bx + c 的图像始终是一条抛物线。如果 a > 0，抛物线开口向上，形似”U”。如果 a < 0，抛物线开口向下，形似倒置的"U"。

The points where the graph crosses the x-axis are exactly the roots of the equation ax² + bx + c = 0. Therefore, the discriminant tells you how many times the graph intersects the x-axis:

图像与 x 轴的交点正好是方程 ax² + bx + c = 0 的根。因此，判别式告诉你图像与 x 轴相交多少次：

• Δ > 0: two x-intercepts | 两个 x 轴交点
• Δ = 0: one x-intercept (touches the axis) | 一个 x 轴交点（与轴相切）
• Δ < 0: no x-intercept | 没有 x 轴交点

When sketching, always label the roots, the y-intercept (which is c) and the turning point. These labels show the examiner that you understand the connection between algebra and graphs.

画草图时，务必标出根、y 轴截距（即 c）以及顶点。这些标注能向考官表明你理解代数与图像之间的联系。

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9. Axis of Symmetry and Turning Point | 对称轴与顶点

Every parabola has an axis of symmetry. For y = ax² + bx + c, this vertical line is given by:

每条抛物线都有一条对称轴。对于 y = ax² + bx + c，这条竖直线为：

x = −b / 2a

The turning point (vertex) lies on this axis. Its x-coordinate is x = −b/2a. To find the y-coordinate, substitute this value back into the original equation.

顶点（转折点）位于这条对称轴上。其 x 坐标为 x = −b/2a。要计算 y 坐标，请将该值代回原方程。

For example, y = x² − 4x + 3 has axis of symmetry x = 4/2 = 2. Substituting x = 2 gives y = 4 − 8 + 3 = −1. So the vertex is (2, −1).

例如，y = x² − 4x + 3 的对称轴为 x = 4/2 = 2。代入 x = 2 得 y = 4 − 8 + 3 = −1。因此顶点为 (2, −1)。

Notice that completing the square gives the same result immediately: y = (x − 2)² − 1, so the vertex is at (2, −1). Use the method that feels most reliable to you.

注意，配方法也能立即得到相同结果：y = (x − 2)² − 1，所以顶点在 (2, −1)。选择你最有把握的方法。

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

Quadratics often appear in real-world problems involving areas, motion and numbers. Always define the unknown variable first, translate the situation into an equation, and then solve.

二次方程常出现在涉及面积、运动和数字的实际问题中。务必先设定未知数，将题意转化为方程，然后求解。

Example: A rectangle has length 3 cm longer than its width, and its area is 40 cm². Let the width be x. Then the length is x + 3, and the area equation is:

示例：一个长方形的长比宽长 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3，面积方程为：

x(x + 3) = 40

Expanding gives x² + 3x − 40 = 0. Factorising: (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since a length cannot be negative, the width is 5 cm and the length is 8 cm.

展开得 x² + 3x − 40 = 0。因式分解：(x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于长度不能为负数，宽为 5 cm，长为 8 cm。

Always reject solutions that do not make sense in the context. In some exam questions, both solutions may be valid; read the question carefully to decide.

始终排除在情境中不合理的解。在某些考试题中，两个解都可能有效；请仔细阅读题目来判断。

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11. Common Mistakes to Avoid | 常见错误提醒

Even strong students lose marks on quadratics because of small errors. Here are the most frequent mistakes and how to avoid them:

即使是优秀的学生也会因为一些小错误在二次方程上失分。以下是最常见的错误及避免方法：

• Not rearranging to standard form | 未整理成标准形式: Always bring all terms to one side before factorising or using the formula.
• Forgetting a ≠ 0 | 忘记 a ≠ 0: If a = 0, the equation is linear, not quadratic.
• Sign errors with negative numbers | 负数符号错误: In the formula, substitute carefully. For b = −5, write −(−5) = +5.
• Incomplete factorisation | 因式分解不完整: Always check that expanding your brackets recovers the original expression.
• Losing one solution | 丢失一个解: When solving (x − 3)(2x + 1) = 0, remember to set both brackets to zero.

After solving, check your answers by substituting them back into the original equation. This quick check can catch many careless errors.

解完后，将答案代回原方程进行验算。这种快速检查能发现许多粗心错误。

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12. Choosing the Best Method | 选择最佳解法

In an exam, you need to choose the most efficient method for each question. The table below summarises the options:

在考试中，你需要为每道题选择最有效的方法。下表总结了各种选择：

Method | 方法	Best used when | 适用情况
Factorisation | 因式分解	The expression factors neatly and quickly | 表达式可快速整齐分解
Quadratic formula | 求根公式	Factorisation is difficult or impossible | 难以或无法因式分解
Completing the square | 配方法	You need the vertex or exact surd answers | 需要顶点或精确根式答案

Remember the discriminant: if b² − 4ac is not a perfect square, factorisation will not give simple answers, and the formula is usually safer. With regular practice, you will learn to recognise the best strategy instantly.

记住判别式：如果 b² − 4ac 不是完全平方数，因式分解不会给出简单答案，此时使用公式通常更稳妥。通过定期练习，你将学会瞬间识别最佳策略。

Quadratic equations are a core skill in IGCSE Mathematics. Master factorisation, the formula and the discriminant, and always check your working. With practice, these problems will become fast and reliable marks for you.

一元二次方程是 IGCSE 数学的核心技能。掌握因式分解、求根公式和判别式，并始终检查你的计算过程。通过练习，这些题目将为你带来快速而稳定的分数。

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