Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, graphs, coordinate geometry, and even in problem-solving questions. Mastering them is essential for achieving a high grade.

二次方程是 IGCSE 数学大纲中最重要的主题之一。它出现在代数、函数图像、坐标几何乃至应用题中。掌握二次方程是取得高分的关键。

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

A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\) and \(c\) are constants and \(a \neq 0\). The highest power of the variable is 2, which is why it is called ‘quadratic’ (from the Latin ‘quadratus’, meaning square).

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量的最高次数是 2，因此被称为“二次”（quadratic 源自拉丁语 quadratus，意为“平方”）。

Examples of quadratic equations include:

二次方程的例子包括：

• x² – 4x + 3 = 0
• 2x² + 7x – 15 = 0
• x² = 9 (which can be rewritten as x² – 9 = 0)

It is important to note that if \(a = 0\), the equation becomes linear, not quadratic, because the x² term disappears.

要注意，如果 a = 0，方程就会变成一次方程而非二次方程，因为 x² 项消失了。

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2. Expanding and Factorising | 展开与因式分解

Before solving quadratic equations, you must be confident with two algebraic skills: expanding brackets and factorising. These are inverse operations.

在解二次方程之前，你必须熟练掌握两种代数技能：去括号展开和因式分解。这两种运算是互逆的。

When expanding, you multiply every term in the first bracket by every term in the second bracket. This is often remembered as FOIL (First, Outer, Inner, Last).

展开时，你需要将第一个括号中的每一项与第二个括号中的每一项相乘。这通常用 FOIL 法则来记忆（First 首项、Outer 外项、Inner 内项、Last 末项）。

(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15

Factorising is the reverse process. To factorise x² + 8x + 15, we look for two numbers that multiply to give 15 and add to give 8. The numbers 3 and 5 work, so:

因式分解是相反的过程。要将 x² + 8x + 15 因式分解，我们需要找两个数，它们相乘得 15，相加得 8。数字 3 和 5 满足条件，因此：

x² + 8x + 15 = (x + 3)(x + 5)

For quadratics with a coefficient of x² greater than 1, such as 2x² + 7x + 3, you can use the ‘ac method’ or ‘trial and error’ to find the correct bracket arrangement.

对于 x² 项系数大于 1 的二次式，例如 2x² + 7x + 3，可以使用“ac 方法”或“试错法”来找到正确的括号排列。

2x² + 7x + 3 = (2x + 1)(x + 3)

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3. Solving by Factorising | 用因式分解法求解

Once a quadratic equation is factorised, solving it becomes straightforward. The key principle is the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

一旦二次方程完成因式分解，求解就变得简单了。关键原理是零积性质：如果两个表达式的乘积为零，那么至少其中一个必须为零。

Consider the equation x² – 5x + 6 = 0. First, factorise the left-hand side:

考虑方程 x² – 5x + 6 = 0。首先对左边进行因式分解：

x² – 5x + 6 = (x – 2)(x – 3) = 0

Since the product is zero, either x – 2 = 0 or x – 3 = 0. Therefore:

由于乘积为零，所以要么 x – 2 = 0，要么 x – 3 = 0。因此：

x = 2 or x = 3

Always check your solutions by substituting them back into the original equation. For x = 2: 4 – 10 + 6 = 0. For x = 3: 9 – 15 + 6 = 0. Both check out.

务必通过将解代回原方程来检验。对于 x = 2：4 – 10 + 6 = 0。对于 x = 3：9 – 15 + 6 = 0。两个解都成立。

Remember: a quadratic equation can have two distinct solutions, one repeated solution, or no real solutions.

记住：二次方程可以有两个不同的解、一个重根，或者没有实数解。

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4. Solving by Completing the Square | 用配方法求解

Completing the square is a method that works for any quadratic equation, even those that cannot be factorised easily. The idea is to rewrite the equation in the form (x + p)² + q = 0.

配方法适用于任何二次方程，即使那些不容易因式分解的方程也适用。其思想是将方程改写为 (x + p)² + q = 0 的形式。

Take the equation x² + 6x + 2 = 0. To complete the square, we halve the coefficient of x, which is 6, giving 3. We then write:

以方程 x² + 6x + 2 = 0 为例。配方时，将 x 的系数 6 除以 2，得到 3。然后写成：

x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7

So the equation becomes (x + 3)² – 7 = 0. Now solve:

因此方程变为 (x + 3)² – 7 = 0。现在求解：

(x + 3)² = 7
x + 3 = ±√7
x = -3 ± √7

This gives the two solutions x = -3 + √7 and x = -3 – √7. Note that completing the square naturally introduces the ± symbol.

这样得到两个解 x = -3 + √7 和 x = -3 – √7。注意配方过程中自然引入了 ± 符号。

For quadratics where the coefficient of x² is not 1, divide through by that coefficient first. For example, 2x² + 8x + 3 = 0 becomes x² + 4x + 1.5 = 0 before completing the square.

对于 x² 项系数不为 1 的二次式，先除以该系数。例如，2x² + 8x + 3 = 0 先化为 x² + 4x + 1.5 = 0 再进行配方。

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5. The Quadratic Formula | 二次求根公式

The quadratic formula is a universal tool for solving any quadratic equation. 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工具。对于 ax² + bx + c = 0，解由下式给出：

x = (-b ± √(b² – 4ac)) / 2a

This formula is on the IGCSE formula sheet, but you must know how to apply it correctly. Let us solve 2x² + 5x – 3 = 0 using the formula.

这个公式会出现在 IGCSE 公式表中，但你必须知道如何正确应用。让我们用公式求解 2x² + 5x – 3 = 0。

Here, a = 2, b = 5, c = -3. Substituting into the formula:

这里 a = 2，b = 5，c = -3。代入公式：

x = (-5 ± √(5² – 4 × 2 × (-3))) / (2 × 2)
x = (-5 ± √(25 + 24)) / 4
x = (-5 ± √49) / 4
x = (-5 ± 7) / 4

This gives x = (-5 + 7) / 4 = 2/4 = 0.5 and x = (-5 – 7) / 4 = -12/4 = -3. The two solutions are x = 0.5 and x = -3.

因此 x = (-5 + 7) / 4 = 2/4 = 0.5，以及 x = (-5 – 7) / 4 = -12/4 = -3。两个解为 x = 0.5 和 x = -3。

When using the quadratic formula, take extra care with negative signs, especially when c is negative. A common mistake is to incorrectly compute b² – 4ac when c is negative.

使用二次求根公式时，要特别注意负号，尤其是 c 为负数时。常见的错误是在 c 为负时错误地计算 b² – 4ac。

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6. The Discriminant | 判别式

The expression b² – 4ac inside the quadratic formula is called the discriminant, often denoted by Δ (the Greek letter ‘delta’). It tells us the nature of the roots without fully solving the equation.

二次求根公式中的表达式 b² – 4ac 称为判别式，通常用 Δ（希腊字母“德尔塔”）表示。无需完全求解方程，它就能告诉我们根的性质。

• If Δ > 0, there are two distinct real roots.
• 如果 Δ > 0，方程有两个不同的实数根。
• If Δ = 0, there is exactly one repeated real root.
• 如果 Δ = 0，方程有一个重根（两个相等的实数根）。
• If Δ < 0, there are no real roots (the graph does not cross the x-axis).
• 如果 Δ < 0，方程没有实数根（图像不穿过 x 轴）。

For example, consider x² + 2x + 5 = 0. Here Δ = 2² – 4 × 1 × 5 = 4 – 20 = -16. Since Δ < 0, this equation has no real solutions.

例如，考虑 x² + 2x + 5 = 0。这里 Δ = 2² – 4 × 1 × 5 = 4 – 20 = -16。由于 Δ < 0，该方程没有实数解。

Exam questions often ask you to determine the number of roots for a given quadratic, or to find the value of a parameter that gives a certain number of roots. These questions test your understanding of the discriminant.

考试题经常要求你判断给定二次方程的根的个数，或求使方程有特定根个数的参数值。这类题考查你对判别式的理解。

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

The graph of a quadratic function y = ax² + bx + c is a parabola. The sign of a determines the direction of the curve: if a > 0, the parabola opens upwards (like a smile); if a < 0, it opens downwards (like a frown).

二次函数 y = ax² + bx + c 的图像是抛物线。a 的符号决定曲线的开口方向：如果 a > 0，抛物线开口向上（像微笑）；如果 a < 0，抛物线开口向下（像皱眉）。

Key features of a quadratic graph include:

二次函数图像的关键特征包括：

• The line of symmetry, which passes through the vertex: x = -b / 2a
• 对称轴，穿过顶点：x = -b / 2a
• The vertex (turning point), which is a maximum if a < 0 and a minimum if a > 0
• 顶点（转向点），当 a < 0 时为最大值点，当 a > 0 时为最小值点
• The y-intercept at (0, c)
• y 轴截距为 (0, c)
• The x-intercepts, which correspond to the solutions of ax² + bx + c = 0
• x 轴截距，对应 ax² + bx + c = 0 的解

Axis of symmetry: x = -b / 2a   |   Vertex: (-b / 2a, c – b² / 4a)

For example, the graph of y = x² – 4x + 3 has a line of symmetry at x = 2, and it crosses the x-axis at x = 1 and x = 3 (since x² – 4x + 3 = (x – 1)(x – 3)).

例如，y = x² – 4x + 3 的图像对称轴为 x = 2，与 x 轴交于 x = 1 和 x = 3（因为 x² – 4x + 3 = (x – 1)(x – 3)）。

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8. Solving Quadratics by Graphing | 用图像法解二次方程

You can solve a quadratic equation by drawing its graph and reading off the x-intercepts. At each x-intercept, y = 0, so the x-coordinates of these points are the roots of the equation.

你可以通过画出二次函数的图像并读取 x 轴截距来解二次方程。在每个 x 轴截距处，y = 0，因此这些点的 x 坐标就是方程的根。

If the graph is given in a question, you may be asked to solve a related equation such as ax² + bx + c = k. To do this, draw the horizontal line y = k on the graph and read the x-coordinates where it crosses the parabola.

如果题目中给出了图像，可能会要求你解相关方程，如 ax² + bx + c = k。做法是在图像上画水平线 y = k，然后读取它与抛物线交点的 x 坐标。

Graphical methods are less precise than algebraic methods, but they are useful for estimating solutions and for understanding the relationship between equations and graphs.

图像法不如代数方法精确，但对于估算解以及理解方程与图像之间的关系非常有用。

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9. Applications and Word Problems | 应用与文字题

Quadratic equations appear in many real-world contexts, including projectile motion, area problems, and number puzzles. IGCSE exams frequently include word problems that lead to a quadratic equation.

二次方程出现在许多实际情境中，包括抛体运动、面积问题和数字谜题。IGCSE 考试经常包含能转化为二次方程的应用题。

Consider this classic problem: The length of a rectangle is 3 cm more than its width, and its area is 40 cm². Find the dimensions of the rectangle.

考虑这个经典问题：一个矩形的长比宽多 3 cm，面积为 40 cm²。求矩形的尺寸。

Let the width be x cm. Then the length is (x + 3) cm. Area = length × width, so:

设宽为 x cm，则长为 (x + 3) cm。面积 = 长 × 宽，因此：

x(x + 3) = 40
x² + 3x – 40 = 0
(x + 8)(x – 5) = 0

So x = -8 or x = 5. Since a length cannot be negative, we reject x = -8. The width is 5 cm and the length is 8 cm.

因此 x = -8 或 x = 5。由于长度不能为负，我们舍去 x = -8。宽为 5 cm，长为 8 cm。

When solving word problems, always state what your variable represents, form the equation carefully, solve it, and then check whether each solution makes sense in the context of the problem.

解应用题时，务必说明变量代表什么，仔细建立方程，求解后再检查每个解在问题情境中是否有意义。

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10. Common Mistakes and Exam Tips | 常见错误与应试技巧

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

许多学生在二次方程题目上因为可以避免的错误而失分。以下是最常见的陷阱以及如何避免它们。

• Forgetting to rearrange: Always write the equation in the form ax² + bx + c = 0 before factorising or using the formula.
• 忘记移项：在因式分解或使用公式之前，务必把方程写成 ax² + bx + c = 0 的形式。
• Sign errors: When substituting into the quadratic formula, carefully handle negative values of b and c.
• 符号错误：代入二次求根公式时，小心处理 b 和 c 的负值。
• Losing solutions: When taking square roots, always write ±. For example, x² = 9 gives x = ±3, not x = 3 only.
• 漏解：开平方时，务必写 ±。例如，x² = 9 的解是 x = ±3，而不仅仅是 x = 3。
• Not checking answers: Substitute your solutions back into the original equation to verify they are correct.
• 不检验答案：把解代回原方程，验证其正确性。

Also remember to show all your working. In IGCSE exams, method marks are awarded even if your final answer is incorrect due to a small arithmetic slip.

还要记住展示完整的解题过程。在 IGCSE 考试中，即使最终答案因小幅算术失误而错误，仍然会获得方法分。

Finally, practise factorising mentally with simple quadratics, as this will save valuable time in the exam. Quick mental factorisation helps you decide which method to use.

最后，练习对简单二次式的快速因式分解，这将在考试中节省宝贵时间。快速心算因式分解有助于你决定使用哪种方法。

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11. Summary | 总结

Quadratic equations can be solved using four main methods: factorising, completing the square, the quadratic formula, and graphical methods. Each method has its advantages.

二次方程可以通过四种主要方法求解：因式分解、配方法、二次求根公式和图像法。每种方法都有其优势。

Method | 方法	Best Used When | 适用情况
Factorising | 因式分解	The equation factorises neatly with integers | 方程能轻松用整数因式分解
Completing the square | 配方法	Finding the vertex or solving when factorising is difficult | 需要求顶点，或因式分解困难时
Quadratic formula | 二次求根公式	Any quadratic equation; reliable when coefficients are messy | 适用于任何二次方程；系数复杂时很可靠
Graphing | 图像法	Estimating roots or solving related equations from a given curve | 估算根，或从给定图像解相关方程

Whichever method you choose, always check the discriminant to understand the nature of the roots and verify your final answers by substitution.

无论选择哪种方法，都要通过判别式了解根的性质，并通过代入来验证最终答案。

With regular practice, solving quadratic equations will become second nature. Keep your working organised, watch out for sign errors, and always present your final answer clearly.

通过定期练习，解二次方程将变得得心应手。保持解题过程条理清晰，留意符号错误，并清晰地呈现最终答案。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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