Solving Quadratic Equations: Factorisation, Formula, and Graphs | 解二次方程：因式分解、公式法与图像

📚 Solving Quadratic Equations: Factorisation, Formula, and Graphs | 解二次方程：因式分解、公式法与图像

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in almost every exam paper, either as a direct question or as part of a larger problem involving graphs, areas, or motion. Understanding how to solve quadratic equations by factorisation, by using the quadratic formula, and by completing the square is essential for achieving a high grade.

二次方程是 IGCSE 数学中最重要的话题之一。它几乎出现在每一份试卷中，既可能作为直接考题出现，也可能作为图像、面积或运动等更大问题的一部分。掌握通过因式分解、二次求根公式和配方法解二次方程，是取得高分的关键。

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1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The reason we require a ≠ 0 is that if a = 0, the equation becomes linear, not quadratic. The highest power of the variable x is 2, which is why it is called “quadratic”.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数，且 a ≠ 0。我们要求 a ≠ 0 的原因是，如果 a = 0，方程就变成了一次方程而不是二次方程。变量 x 的最高次数为 2，因此称为”二次”。

For example, x² + 5x + 6 = 0, 2x² − 3x − 2 = 0, and even equations like x² = 9 can be rearranged into the standard form. In the last case, we rewrite it as x² − 9 = 0, which fits the pattern with a = 1, b = 0, c = −9.

例如，x² + 5x + 6 = 0、2x² − 3x − 2 = 0，甚至像 x² = 9 这样的方程也可以整理成标准形式。对于最后一个例子，我们将其改写为 x² − 9 = 0，此时 a = 1，b = 0，c = −9。

Before solving any quadratic equation, always check that it is in the form ax² + bx + c = 0. If it is not, rearrange it first. This is the single most important preparatory step and it prevents many common errors.

在解任何二次方程之前，务必检查它是否为 ax² + bx + c = 0 的形式。如果不是，先移项整理。这是最重要的准备步骤，可以避免许多常见错误。

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2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method for solving quadratic equations, provided the equation has simple integer roots. The method relies on rewriting the quadratic expression as a product of two linear factors.

因式分解法通常是解二次方程最快的方法，前提是方程具有简单的整数根。该方法的核心是将二次表达式改写为两个一次因式的乘积。

Consider the equation x² + 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give 5. Those numbers are 2 and 3. Therefore we can write:

考虑方程 x² + 5x + 6 = 0。我们需要找到两个数，它们相乘等于 6，相加等于 5。这两个数是 2 和 3。因此我们可以写出：

x² + 5x + 6 = (x + 2)(x + 3) = 0

If the product of two expressions is zero, then at least one of them must be zero. This gives x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3. These are called the roots or solutions of the equation.

如果两个表达式的乘积为零，那么其中至少有一个必须为零。由此可得 x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。这两个值称为方程的根或解。

When the coefficient of x² is greater than 1, for example 2x² − 3x − 2 = 0, we use a slightly different approach. We multiply a and c: 2 × (−2) = −4. We then find two numbers that multiply to −4 and add to −3. These numbers are −4 and 1. We split the middle term:

当 x² 的系数大于 1 时，例如 2x² − 3x − 2 = 0，我们需要使用略有不同的方法。我们将 a 和 c 相乘：2 × (−2) = −4。然后找到两个数相乘等于 −4、相加等于 −3。这两个数是 −4 和 1。我们拆分中间项：

2x² − 4x + x − 2 = 0

Now we factor by grouping: 2x(x − 2) + 1(x − 2) = 0, which gives (2x + 1)(x − 2) = 0. Hence x = −½ or x = 2.

然后我们分组提取公因式：2x(x − 2) + 1(x − 2) = 0，得到 (2x + 1)(x − 2) = 0。因此 x = −½ 或 x = 2。

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3. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that works for any quadratic equation. It also provides the foundation for deriving the quadratic formula and for identifying the vertex of a parabola.

配方法是一种强大的技巧，适用于任何二次方程。它也是推导二次求根公式以及确定抛物线顶点的基础。

Consider x² + 6x + 2 = 0. We start by halving the coefficient of x, which is 6. Half of 6 is 3, and 3² = 9. We rewrite the equation as:

考虑 x² + 6x + 2 = 0。我们先将 x 的系数 6 取半。6 的一半是 3，3² = 9。我们将方程改写为：

(x + 3)² − 9 + 2 = 0

This simplifies to (x + 3)² − 7 = 0. Adding 7 to both sides gives (x + 3)² = 7. Taking the square root of both sides gives x + 3 = ±√7, so x = −3 ± √7. The symbol ± means we have two solutions: x = −3 + √7 or x = −3 − √7.

化简后得到 (x + 3)² − 7 = 0。两边同时加 7，得到 (x + 3)² = 7。两边开平方，得到 x + 3 = ±√7，所以 x = −3 ± √7。符号 ± 表示我们有两个解：x = −3 + √7 或 x = −3 − √7。

If the coefficient of x² is not 1, divide every term by that coefficient before completing the square. For example, 2x² + 8x + 5 = 0 becomes x² + 4x + 2.5 = 0 first, and then we proceed as above.

如果 x² 的系数不是 1，则先将每一项除以该系数，然后再配方。例如，2x² + 8x + 5 = 0 首先变为 x² + 4x + 2.5 = 0，然后按上述步骤进行。

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4. The Quadratic Formula | 二次求根公式

The quadratic formula is a universal method for solving any quadratic equation. It is especially useful when the equation cannot be factorised easily. The formula states that for ax² + bx + c = 0, the solutions are given by:

二次求根公式是解任何二次方程的通用方法。当方程不易因式分解时，它尤其有用。公式指出，对于 ax² + bx + c = 0，其解为：

x = (−b ± √(b² − 4ac)) ÷ 2a

Let us apply this to 2x² − 3x − 2 = 0, where a = 2, b = −3, c = −2. We substitute these values:

我们将此公式应用于 2x² − 3x − 2 = 0，其中 a = 2，b = −3，c = −2。代入这些值：

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

This simplifies to x = (3 ± √(9 + 16)) ÷ 4, which is x = (3 ± √25) ÷ 4. Since √25 = 5, we get x = (3 + 5) ÷ 4 = 2 or x = (3 − 5) ÷ 4 = −½. These match the factorisation result, as expected.

这化简为 x = (3 ± √(9 + 16)) ÷ 4，即 x = (3 ± √25) ÷ 4。由于 √25 = 5，我们得到 x = (3 + 5) ÷ 4 = 2 或 x = (3 − 5) ÷ 4 = −½。这与因式分解的结果一致，符合预期。

When using the quadratic formula, be very careful with negative signs. It is recommended to write down the values of a, b and c before substituting, and to use brackets where necessary.

使用二次求根公式时，要特别注意负号。建议在代入之前先写下 a、b 和 c 的值，并在需要时使用括号。

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5. The Discriminant | 判别式

The expression b² − 4ac in the quadratic formula is called the discriminant. It tells us how many real roots a quadratic equation has, without solving the equation fully.

二次求根公式中的 b² − 4ac 这一表达式称为判别式。它可以告诉我们二次方程有多少个实数根，而无需完整解方程。

Discriminant value	Number of real roots	Graph interpretation
b² − 4ac > 0	Two distinct real roots	Parabola crosses the x-axis twice
b² − 4ac = 0	Exactly one real root (repeated)	Parabola touches the x-axis once
b² − 4ac < 0	No real roots	Parabola does not intersect the x-axis

For the equation x² + 6x + 9 = 0, the discriminant is 6² − 4 × 1 × 9 = 36 − 36 = 0. This means the equation has a repeated root. Indeed, x² + 6x + 9 = (x + 3)², so x = −3 is the only solution.

对于方程 x² + 6x + 9 = 0，判别式为 6² − 4 × 1 × 9 = 36 − 36 = 0。这意味着方程有一个重根。事实上，x² + 6x + 9 = (x + 3)²，所以 x = −3 是唯一解。

For x² + x + 5 = 0, the discriminant is 1² − 4 × 1 × 5 = 1 − 20 = −19, which is negative. Therefore the equation has no real solutions. This is why we say the graph of y = x² + x + 5 never touches the x-axis.

对于 x² + x + 5 = 0，判别式为 1² − 4 × 1 × 5 = 1 − 20 = −19，是负数。因此方程没有实数解。这就是为什么我们说 y = x² + x + 5 的图像永远不会触及 x 轴。

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6. Word Problems with Quadratic Equations | 二次方程应用题

Quadratic equations often appear in word problems involving areas, consecutive numbers, or projectile motion. The key is to translate the words into an equation, solve it, and then check which solutions make sense in the context.

二次方程经常出现在涉及面积、连续整数或抛体运动的应用题中。关键是将文字转化为方程，求解，然后检查哪些解在上下文中是合理的。

Example: The area of a rectangle is 36 cm². Its length is 5 cm longer than its width. Find the dimensions of the rectangle.

例：一个矩形的面积为 36 cm²。它的长比宽长 5 cm。求该矩形的尺寸。

Let the width be x cm. Then the length is x + 5 cm. The area is x(x + 5) = 36, which expands to x² + 5x = 36. Rearranging gives x² + 5x − 36 = 0. Factorising: (x + 9)(x − 4) = 0, so x = −9 or x = 4. Since a width cannot be negative, we discard x = −9. The width is 4 cm and the length is 9 cm.

设宽为 x cm，则长为 x + 5 cm。面积为 x(x + 5) = 36，展开得 x² + 5x = 36。移项得到 x² + 5x − 36 = 0。因式分解：(x + 9)(x − 4) = 0，所以 x = −9 或 x = 4。由于宽度不能为负数，我们舍弃 x = −9。宽度为 4 cm，长度为 9 cm。

In such problems, always consider whether both roots are valid. Negative lengths, negative times, or fractional numbers of people are usually impossible. Reading the problem carefully and writing a concluding sentence with units will earn full marks.

在解决此类问题时，始终要考虑两个根是否都有意义。负的长度、负的时间或分数的人数通常是不可能的。仔细阅读题目，并用带单位的一句话写出结论，才能拿到满分。

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7. Graphs of Quadratic Functions | 二次函数的图像

The graph of a quadratic function y = ax² + bx + c is a curve called a parabola. If a > 0, the parabola opens upwards like a “U” shape. If a < 0, it opens downwards like an upside-down "U".

二次函数 y = ax² + bx + c 的图像是一条称为抛物线的曲线。如果 a > 0，抛物线开口向上，呈”U”形；如果 a < 0，抛物线开口向下，呈倒"U"形。

The vertex (turning point) of the parabola can be found by completing the square. For example, y = x² − 4x + 1 can be written as y = (x − 2)² − 3. This tells us that the vertex is at (2, −3). The line of symmetry is the vertical line x = 2.

抛物线的顶点（转向点）可以通过配方法找到。例如，y = x² − 4x + 1 可以写成 y = (x − 2)² − 3。这告诉我们顶点在 (2, −3) 处。对称轴是竖直线 x = 2。

The y-intercept of the graph is simply the value of c, because when x = 0, y = c. For the equation above, the y-intercept is at (0, 1). The x-intercepts are the roots of the quadratic equation, which we find using any of the methods discussed earlier.

图像的 y 截距就是 c 的值，因为当 x = 0 时，y = c。对于上述方程，y 截距在 (0, 1) 处。x 截距是二次方程的根，可以用前面讨论的任何一种方法求出。

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8. Solving Quadratic Equations by Graphical Methods | 图像法求解二次方程

Another approach to solving x² + Ax + B = 0 is to plot the graph of y = x² + Ax + B and read off the x-coordinates where the curve crosses the x-axis. These crossing points are exactly the roots of the equation.

另一种求解 x² + Ax + B = 0 的方法是绘制 y = x² + Ax + B 的图像，然后读出曲线与 x 轴交点的 x 坐标。这些交点正是方程的根。

Sometimes a question may ask you to solve x² − 2x − 3 = 0 given the graph of y = x² − 2x − 3. Reading from the graph, the curve crosses the x-axis at x = −1 and x = 3, so those are the solutions.

有时题目会给出 y = x² − 2x − 3 的图像，要求解 x² − 2x − 3 = 0。从图像上读出，曲线在 x = −1 和 x = 3 处与 x 轴相交，因此这两个值就是解。

Graphical methods are less precise than algebraic methods unless the roots are integers. However, they are useful for understanding the relationship between the equation and its graph. For non-integer roots, you may be asked to give approximate answers, often to one decimal place.

图像法没有代数方法精确，除非根是整数。然而，它们有助于理解方程与其图像之间的关系。对于非整数根，题目可能要求给出近似答案，通常精确到一位小数。

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

One of the most common mistakes is forgetting to rearrange the equation into the form ax² + bx + c = 0 before factorising. Another frequent error is misapplying the square root property: for example, writing x² = 9 as x = 3 instead of x = ±3.

最常见的错误之一是忘记在因式分解之前将方程整理为 ax² + bx + c = 0 的形式。另一个常见错误是错误地应用平方根性质：例如，将 x² = 9 写成 x = 3 而不是 x = ±3。

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

  始终将解代回原方程进行验算。

• When using the quadratic formula, ensure you divide the entire numerator by 2a, not just one part of it.

  使用二次求根公式时，确保整个分子除以 2a，而不是只除以其中一部分。

• If a quadratic equation has no constant term, such as x² − 4x = 0, factorise directly: x(x − 4) = 0.

  如果二次方程没有常数项，例如 x² − 4x = 0，可以直接因式分解：x(x − 4) = 0。

• For equations like (x − 1)² = 16, take the square root of both sides: x − 1 = ±4, so x = 5 or x = −3.

  对于形如 (x − 1)² = 16 的方程，两边开平方：x − 1 = ±4，所以 x = 5 或 x = −3。

In examinations, show full working. The method mark is often awarded even if your final answer is incorrect, but only if your working is clear and logical. Use a consistent method that you are comfortable with, and practise it until it becomes automatic.

考试中要展示完整的解题过程。即使最终答案不正确，只要过程清晰合理，通常也能获得方法分。选择一种你熟悉且一致的方法，并反复练习直到掌握。

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10. Summary and Practice Structure | 总结与练习框架

In this article we have covered the four main ways to solve a quadratic equation: factorisation, completing the square, the quadratic formula, and graphical methods. We also explored the discriminant and its significance in determining the nature of the roots.

在本文中，我们涵盖了求解二次方程的四种主要方法：因式分解、配方法、二次求根公式和图像法。我们还探讨了判别式及其在确定根的性质方面的重要意义。

A good revision strategy is to solve at least five quadratic equations per session, using a different method each time, and to check your work by substituting the answers back. This will build both speed and confidence for the exam.

一个好的复习策略是每次练习至少解五道二次方程，每次都使用不同的方法，并通过回代答案来检验。这将有助于提高考试时的速度和信心。

Remember that many IGCSE problems combine quadratics with other topics such as simultaneous equations, inequalities, and sequences. Mastering quadratics thoroughly will give you a strong foundation across the whole syllabus.

请记住，许多 IGCSE 考题将二次方程与联立方程、不等式和数列等主题相结合。彻底掌握二次方程将为你学习整个大纲打下坚实的基础。

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