Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is a polynomial equation of degree 2, typically written in the form ax² + bx + c = 0, where a ≠ 0. Solving such equations is a core skill in the Edexcel IGCSE Mathematics syllabus. This article explains the key methods, the discriminant, and real-world applications, with paired English and Chinese explanations.

二次方程是最高次数为 2 的多项式方程，通常写成 ax² + bx + c = 0 的形式，其中 a ≠ 0。解二次方程是 Edexcel IGCSE 数学大纲中的核心技能。本文将讲解主要方法、判别式以及实际应用，并配以中英双语对照说明。

1. Standard Form and Key Vocabulary | 标准形式与关键词汇

A quadratic equation must contain an x² term. The standard form is ax² + bx + c = 0, where a, b, and c are constants, and a is not zero. If a = 0, the equation becomes linear.

二次方程必须含有 x² 项。其标准形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不为零。如果 a = 0，方程就变成了一次方程。

Roots / solutions: values of x that make the equation true.
根 / 解：使方程成立的 x 的值。
Leading coefficient a: the coefficient of x².
首项系数 a：x² 的系数。
Constant term c: the term without x.
常数项 c：不含 x 的项。

2. Method 1: Factorisation | 方法一：因式分解法

Factorisation is often the quickest method when the quadratic can be written as a product of two binomials. For example, x² − 5x + 6 = 0 can be factored as (x − 2)(x − 3) = 0. Then, by the zero product property, either x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

当二次方程可以写成两个二项式的乘积时，因式分解法通常是最快捷的方法。例如，x² − 5x + 6 = 0 可以分解为 (x − 2)(x − 3) = 0。根据零积性质，要么 x − 2 = 0，要么 x − 3 = 0，所以 x = 2 或 x = 3。

To factorise ax² + bx + c = 0, look for two numbers that multiply to give ac and add to give b. For x² + 7x + 12, we need two numbers multiplying to 12 and adding to 7: 3 and 4. Hence (x + 3)(x + 4) = 0, so x = −3 or x = −4.

要分解 ax² + bx + c = 0，需要找到两个数，它们的乘积等于 ac，和等于 b。对于 x² + 7x + 12，我们需要两个数乘积为 12 且和为 7：即 3 和 4。因此 (x + 3)(x + 4) = 0，所以 x = −3 或 x = −4。

3. Method 2: Completing the Square | 方法二：配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. Start with x² + bx, add and subtract (b/2)², then factor the perfect square trinomial. For example, x² + 6x + 2 = 0 becomes (x + 3)² − 7 = 0.

配方法将二次方程改写为 (x + p)² + q 的形式。先处理 x² + bx，加上并减去 (b/2)²，然后因式分解得到完全平方三项式。例如，x² + 6x + 2 = 0 可化为 (x + 3)² − 7 = 0。

Then solve by isolating the square: (x + 3)² = 7, so x + 3 = ±√7, giving x = −3 ± √7. This method is useful when the quadratic does not factorise easily.

然后通过移项开平方来求解：(x + 3)² = 7，因此 x + 3 = ±√7，得到 x = −3 ± √7。当二次方程不易因式分解时，这种方法非常有用。

For x² + bx, add (b/2)² to complete the square.

对于 x² + bx，加上 (b/2)² 即可配方。

4. Method 3: The Quadratic Formula | 方法三：求根公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is derived by completing the square on the general form.

求根公式可以解任意形如 ax² + bx + c = 0 的二次方程。它通过对一般形式配方推导得出。

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute: x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± (√10)/2.

例如，解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入得：x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± (√10)/2。

5. The Discriminant b² − 4ac | 判别式 b² − 4ac

The expression b² − 4ac under the square root is called the discriminant. It tells us how many real roots a quadratic equation has without solving it fully.

根号下的表达式 b² − 4ac 称为判别式。它无需完整求解即可告诉我们二次方程有多少个实数根。

b² − 4ac > 0: two distinct real roots.
b² − 4ac > 0：两个不相等的实数根。
b² − 4ac = 0: exactly one real root (a repeated root).
b² − 4ac = 0：恰好一个实数根（重根）。
b² − 4ac < 0: no real roots (two complex roots).
b² − 4ac < 0：没有实数根（两个复数根）。

For example, x² − 4x + 4 = 0 has discriminant 16 − 16 = 0, so it has one repeated root: x = 2.

例如，x² − 4x + 4 = 0 的判别式为 16 − 16 = 0，因此它有一个重根：x = 2。

6. Graphical Interpretation | 图像意义

The graph of a quadratic function y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 are the x-coordinates where the parabola crosses the x-axis.

二次函数 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根就是抛物线与 x 轴交点的横坐标。

If the discriminant is positive, the parabola crosses the x-axis at two points.
如果判别式为正，抛物线与 x 轴有两个交点。
If the discriminant is zero, the parabola touches the x-axis at one point (the vertex).
如果判别式为零，抛物线与 x 轴相切于一点（顶点）。
If the discriminant is negative, the parabola does not intersect the x-axis.
如果判别式为负，抛物线与 x 轴没有交点。

7. Solving by Using the Graph | 利用图像求解

You may be asked to estimate roots from a given graph. The roots are where the curve meets the x-axis. For example, if the curve y = x² − x − 6 crosses the x-axis at x = −2 and x = 3, then the solutions to x² − x − 6 = 0 are x = −2 and x = 3.

你可能会被要求从给定图像中估算根。根就是曲线与 x 轴的交点。例如，如果曲线 y = x² − x − 6 与 x 轴交于 x = −2 和 x = 3，那么方程 x² − x − 6 = 0 的解就是 x = −2 和 x = 3。

Sometimes the axis of symmetry can help find the vertex. For y = ax² + bx + c, the x-coordinate of the vertex is x = −b/(2a).

有时对称轴可以帮助找到顶点。对于 y = ax² + bx + c，顶点的横坐标为 x = −b/(2a)。

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

Many real-life problems can be modelled by quadratic equations. For example, the area of a rectangle is 30 cm², and its length is 4 cm more than its width. Let the width be x cm. Then x(x + 4) = 30, so x² + 4x − 30 = 0.

许多实际问题可以用二次方程建模。例如，一个矩形的面积为 30 cm²，长比宽多 4 cm。设宽为 x cm，则 x(x + 4) = 30，即 x² + 4x − 30 = 0。

Solve using the quadratic formula: x = (−4 ± √(16 + 120)) / 2 = (−4 ± √136) / 2 = (−4 ± 2√34) / 2 = −2 ± √34. Since width cannot be negative, x = −2 + √34 ≈ 3.83 cm.

使用求根公式解：x = (−4 ± √(16 + 120)) / 2 = (−4 ± √136) / 2 = (−4 ± 2√34) / 2 = −2 ± √34。由于宽度不能为负，所以 x = −2 + √34 ≈ 3.83 cm。

Always check whether each root is valid in the context of the problem.

始终检查每个根在问题情境中是否合理。

9. Factorising Harder Quadratics (a ≠ 1) | 系数不为 1 的因式分解

When the coefficient of x² is not 1, factorisation requires more care. For example, 2x² + 7x + 3. Multiply a and c: 2 × 3 = 6. Find two numbers whose product is 6 and sum is 7: 1 and 6. Split the middle term: 2x² + 1x + 6x + 3. Then factor by grouping: x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).

当 x² 的系数不为 1 时，因式分解需要更加小心。例如，2x² + 7x + 3。将 a 和 c 相乘：2 × 3 = 6。找到两个数乘积为 6 且和为 7：即 1 和 6。拆分中间项：2x² + 1x + 6x + 3。然后分组分解：x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)。

Therefore, 2x² + 7x + 3 = 0 gives x = −1/2 or x = −3.

因此，2x² + 7x + 3 = 0 的解为 x = −1/2 或 x = −3。

10. The Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of the roots is α + β = −b/a, and the product is αβ = c/a.

对于二次方程 ax² + bx + c = 0，若其根为 α 和 β，则根的和为 α + β = −b/a，根的积为 αβ = c/a。

Example: For x² − 5x + 6 = 0, the roots are 2 and 3. The sum is 5 = −(−5)/1, and the product is 6 = 6/1. This relationship helps check solutions or form a quadratic when roots are known.

例如：对于 x² − 5x + 6 = 0，根为 2 和 3。和为 5 = −(−5)/1，积为 6 = 6/1。利用这种关系可以检验解，或在已知根时构造二次方程。

11. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Students often lose marks on quadratic equations due to small errors. Being aware of these pitfalls can greatly improve accuracy.

学生在二次方程上常常因小错误而失分。了解这些陷阱可以大大提高准确性。

Forgetting to rearrange to ax² + bx + c = 0 first. Always bring all terms to one side before factorising or using the formula.
忘记先将方程整理为 ax² + bx + c = 0。在分解因式或使用公式前，务必把所有项移到一边。
Dropping negative signs in the quadratic formula. Carefully substitute b, including its sign.
在求根公式中丢掉负号。代入 b 时，注意包含其符号。
Dividing both sides by x unnecessarily. This loses the root x = 0 when x is a factor.
随意两边同时除以 x。当 x 是因子时，这样做会丢失 x = 0 这个根。
Ignoring the ± sign. Your calculator may give only one root.
忽略 ± 符号。计算器可能只给出一个根。

12. Practice Checklist | 练习检查清单

To master solving quadratic equations for the Edexcel IGCSE exam, follow this checklist:

为了在 Edexcel IGCSE 考试中掌握解二次方程，请遵循以下检查清单：

Rearrange into standard form.
整理为标准形式。
Try factorisation first when a = 1.
当 a = 1 时，先尝试因式分解。
Use completing the square or the quadratic formula when needed.
需要时使用配方法或求根公式。
Calculate the discriminant to know the nature of roots.
计算判别式以确定根的性质。
Check each solution by substitution.
通过代入验证每个解。
Interpret roots in the context of word problems.
在应用题中解释根的实际意义。

Practice with past paper questions, focusing on speed and accuracy. Quadratic equations appear in many exam sections, so consistent practice is essential.

通过历年真题进行练习，注重速度和准确性。二次方程在考试的许多部分都会出现，因此持续练习至关重要。

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Solving Quadratic Equations | 解二次方程