Quadratic Equations: Solving Methods and Applications | 二次方程：解法与应用

📚 Quadratic Equations: Solving Methods and Applications | 二次方程：解法与应用

Quadratic equations appear frequently in the Edexcel IGCSE Mathematics syllabus. This revision guide explains the essential methods for solving them, the meaning of the discriminant, and how to apply these skills to real exam-style problems.

二次方程在 Edexcel IGCSE 数学考纲中频繁出现。本复习指南讲解解二次方程的基本方法、判别式的意义，以及如何将这些技能应用于典型的考试题型。

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 \(a, b, c\) are constants and \(a \neq 0\). The highest power of the variable is 2, and the graph of such an equation is a parabola.

二次方程是形如 \(ax^2 + bx + c = 0\) 的方程，其中 \(a, b, c\) 为常数，且 \(a \neq 0\)。变量的最高次数是 2，其图像是一条抛物线。

In IGCSE exams, you must be able to identify a quadratic equation and rewrite it in standard form before solving.

在 IGCSE 考试中，你必须能够识别二次方程，并在求解前将其化为标准形式。

2. Standard Form and the Discriminant | 标准形式与判别式

The standard form is \(ax^2 + bx + c = 0\). The discriminant \(D\) is defined as:

标准形式为 \(ax^2 + bx + c = 0\)。判别式 \(D\) 定义为：

D = b² − 4ac

The value of \(D\) tells us how many real roots an equation has:

\(D\) 的值告诉我们方程有多少个实数根：

If \(D > 0\): two distinct real roots.

若 \(D > 0\)：有两个不同的实数根。
If \(D = 0\): exactly one repeated real root.

若 \(D = 0\)：恰好有一个相等的实数根（重根）。
If \(D < 0\): no real roots (two complex roots).

若 \(D < 0\)：没有实数根（有两个复数根）。

3. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic has simple integer coefficients. Write \(ax^2 + bx + c = 0\) as a product of two binomials.

当二次方程的系数为简单的整数时，因式分解往往是最快的方法。将 \(ax^2 + bx + c = 0\) 写成两个二项式的乘积。

Example: Solve \(x^2 – 5x + 6 = 0\).

示例：求解 \(x^2 – 5x + 6 = 0\)。

(x − 2)(x − 3) = 0

Then \(x = 2\) or \(x = 3\).

因此 \(x = 2\) 或 \(x = 3\)。

For equations where \(a \neq 1\), use the product-sum method or the AC method.

当 \(a \neq 1\) 时，可使用乘积-和法或 AC 法。

4. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic as \((x + p)^2 + q\). This method is useful when factorisation is not obvious.

配方法将二次方程改写为 \((x + p)^2 + q\) 的形式。当因式分解不明显时，此方法很有用。

Example: Solve \(x^2 + 6x + 1 = 0\).

示例：求解 \(x^2 + 6x + 1 = 0\)。

Take half of 6, square it, and adjust:

取 6 的一半，平方，然后调整：

(x + 3)² − 9 + 1 = 0 → (x + 3)² = 8

x + 3 = ±√8 → x = −3 ± 2√2

Remember to simplify surds where possible.

记住要尽可能化简根式。

5. The Quadratic Formula | 二次公式

The quadratic formula works for every quadratic equation and is especially useful when factorisation is difficult or impossible.

二次公式适用于所有二次方程，特别在因式分解困难或不可行时非常有用。

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve \(2x^2 − 7x + 3 = 0\). Here \(a = 2, b = −7, c = 3\).

例如，求解 \(2x^2 − 7x + 3 = 0\)。这里 \(a = 2, b = −7, c = 3\)。

x = (7 ± √(49 − 24)) / 4 = (7 ± 5) / 4

So \(x = 3\) or \(x = 0.5\).

因此 \(x = 3\) 或 \(x = 0.5\)。

6. Discriminant and Nature of Roots | 判别式与根的性质

In exams, you may be asked to state whether roots are rational, irrational, or not real. This is controlled by \(D\) and whether \(D\) is a perfect square.

考试中，你可能会被要求说明根是有理数、无理数还是非实数。这取决于 \(D\) 以及 \(D\) 是否为完全平方数。

If \(D > 0\) and \(D\) is a perfect square: two distinct rational roots.

若 \(D > 0\) 且 \(D\) 是完全平方数：两个不同的有理数根。
If \(D > 0\) and \(D\) is not a perfect square: two distinct irrational roots.

若 \(D > 0\) 且 \(D\) 不是完全平方数：两个不同的无理数根。
If \(D = 0\): one repeated rational root.

若 \(D = 0\)：一个相等的有理数根。
If \(D < 0\): no real roots.

若 \(D < 0\)：没有实数根。

Be sure to show the calculation of \(D\) clearly in your working.

在解答中务必清楚展示 \(D\) 的计算过程。

7. Sum and Product of Roots | 根与系数的关系

For a quadratic equation \(ax^2 + bx + c = 0\), if the roots are \(\alpha\) and \(\beta\), then:

对于二次方程 \(ax^2 + bx + c = 0\)，若根为 \(\alpha\) 和 \(\beta\)，则：

α + β = −b / a

αβ = c / a

These relationships allow you to form a new quadratic given the sum and product of its roots.

这些关系允许你在已知根的和与积时构造一个新的二次方程。

Example: Find a quadratic with roots 2 and −5.

示例：求一个以 2 和 −5 为根的二次方程。

Sum = −3, product = −10, so the equation is:

和 = −3，积 = −10，所以方程为：

x² + 3x − 10 = 0

8. Graph of a Quadratic | 二次函数的图像

The graph of \(y = ax^2 + bx + c\) is a parabola. The sign of \(a\) determines the orientation: \(a > 0\) gives a U-shape, and \(a < 0\) gives an upside-down U-shape.

\(y = ax^2 + bx + c\) 的图像是一条抛物线。\(a\) 的符号决定了开口方向：\(a > 0\) 时开口向上，\(a < 0\) 时开口向下。

Key features include the roots (x-intercepts), the y-intercept, and the vertex (turning point). The axis of symmetry is:

关键特征包括根（与x轴交点）、y轴截距和顶点（转向点）。对称轴为：

x = −b / (2a)

The vertex lies on this axis, and its y-coordinate can be found by substitution.

顶点位于该对称轴上，其 y 坐标可通过代入求得。

9. Solving Quadratic Inequalities | 解二次不等式

A common IGCSE extension is solving inequalities such as \(ax^2 + bx + c > 0\) or \(< 0\). First solve the corresponding equation, then test intervals.

IGCSE 常见的延伸题型是解二次不等式，如 \(ax^2 + bx + c > 0\) 或 \(< 0\)。首先解对应的方程，然后测试区间。

Example: Solve \(x^2 – x – 6 > 0\). The roots are −2 and 3. The parabola opens upward, so the inequality is true outside the interval (−2, 3).

示例：解 \(x^2 – x – 6 > 0\)。根为 −2 和 3。抛物线开口向上，因此不等式在区间 (−2, 3) 之外成立。

x < −2 or x > 3

Always draw a quick sketch to avoid sign errors.

务必画简图以避免符号错误。

10. Word Problems and Applications | 应用题与实际应用

Quadratic equations often model area, projectile motion, and profit. Convert the problem into a quadratic equation by defining a variable carefully.

二次方程常被用于建模面积、抛体运动和利润问题。通过仔细定义变量，将问题转化为二次方程。

Example: A rectangle has length 4 cm more than its width. Its area is 45 cm². Find the width.

示例：一个矩形的长比宽多 4 cm，其面积为 45 cm²。求宽。

Let width = \(w\). Then length = \(w + 4\).

设宽为 \(w\)，则长为 \(w + 4\)。

w(w + 4) = 45 → w² + 4w − 45 = 0

Factorising gives \((w + 9)(w − 5) = 0\), so \(w = 5\) cm. Discard the negative solution.

因式分解得 \((w + 9)(w − 5) = 0\)，所以 \(w = 5\) cm。舍去负数解。

11. Common Mistakes and Exam Tips | 常见错误与考试要点

Mistakes often occur when rearranging, forgetting to set the equation to zero, or losing a negative sign. Always check your answers by substitution.

常见错误包括：项移错、忘记将方程化为零、漏写负号。务必通过代回来检查答案。

Always write the equation in the form \(ax^2 + bx + c = 0\) first.

始终先把方程写成 \(ax^2 + bx + c = 0\) 的形式。
When using the formula, be careful with negative signs for \(b\).

使用公式时，注意 \(b\) 的负号。
If a question asks for answers to 2 decimal places, use the quadratic formula or completing the square, not factorisation.

若题目要求答案保留两位小数，应使用二次公式或配方法，而不是因式分解。
Show sufficient working to earn method marks.

写出足够的步骤以获得方法分。

12. Summary | 总结

Mastering quadratics is essential for Edexcel IGCSE Mathematics. Know when to use each method, understand the discriminant, and practise drawing graphs.

掌握二次方程对于 Edexcel IGCSE 数学至关重要。了解何时使用每种方法，理解判别式，并练习画图。

Start with factorisation, then move to completing the square and the quadratic formula. With consistent practice, you will build speed and accuracy.

从因式分解开始，然后掌握配方法和二次公式。通过持续练习，你将提升速度和准确度。

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Quadratic Equations: Solving Methods and Applications | 二次方程：解法与应用