Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in almost every IGCSE Mathematics paper. Understanding how to solve them efficiently is essential for achieving a high grade. This article covers all key methods required by the Edexcel syllabus, with step-by-step explanations and examples.

二次方程几乎出现在每一份 IGCSE 数学试卷中。掌握高效求解二次方程的方法，是取得高分的关键。本文按照 Edexcel 大纲要求，系统讲解所有核心解法，并配逐步例题说明。

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

A quadratic equation is an equation where the highest power of the variable is 2. Its standard form is written as:

二次方程是指变量的最高次数为 2 的方程。其标准形式写成：

ax² + bx + c = 0

Here a, b and c are constants, and a ≠ 0. For example, 2x² − 5x + 3 = 0 is a quadratic equation. The expression ax² + bx + c is called a quadratic expression.

其中 a、b、c 为常数，且 a ≠ 0。例如 2x² − 5x + 3 = 0 就是一个二次方程。表达式 ax² + bx + c 称为二次表达式。

Solving a quadratic equation means finding the values of x that make the equation true. These values are called the roots or solutions. A quadratic equation can have two real roots, one repeated root, or no real roots.

解二次方程就是求出使等式成立的 x 的值。这些值称为方程的根或解。二次方程可能有两个实数根、一个重根，也可能没有实数根。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic expression can be factored into two linear binomials. The key is to use the zero-product property: if the product of two expressions is zero, then at least one of them must be zero.

当二次表达式可以分解为两个一次二项式相乘时，因式分解通常是最快的方法。关键利用零积性质：若两个表达式的乘积为零，则至少其中一个必须为零。

For a quadratic of the form x² + bx + c, we look for two numbers that multiply to give c and add to give b.

对于形如 x² + bx + c 的二次式，我们要找两个数，它们相乘等于 c，相加等于 b。

Example 1: Solve x² − 7x + 10 = 0.

例 1：解方程 x² − 7x + 10 = 0。

We need two numbers that multiply to 10 and add to −7. These are −2 and −5. So:

我们需要找到两个数，它们相乘为 10，相加为 −7。这两个数是 −2 和 −5。因此：

(x − 2)(x − 5) = 0

Then x − 2 = 0 or x − 5 = 0, giving x = 2 or x = 5.

于是 x − 2 = 0 或 x − 5 = 0，解得 x = 2 或 x = 5。

When the coefficient of x² is not 1, factorisation becomes more careful. For example, solve 2x² + 5x − 3 = 0 by factorising as (2x − 1)(x + 3) = 0. Then 2x − 1 = 0 or x + 3 = 0, so x = ½ or x = −3.

当 x² 的系数不为 1 时，因式分解需要更仔细。例如，解 2x² + 5x − 3 = 0，可分解为 (2x − 1)(x + 3) = 0。于是 2x − 1 = 0 或 x + 3 = 0，所以 x = ½ 或 x = −3。

3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form p(x + q)² + r. This method works for any quadratic equation, including those that cannot be factorised easily.

配方法将二次表达式改写为 p(x + q)² + r 的形式。这种方法适用于任何二次方程，包括那些不易因式分解的方程。

Example 2: Solve x² + 6x − 7 = 0 by completing the square.

例 2：用配方法解方程 x² + 6x − 7 = 0。

Start by moving the constant term to the right:

先把常数项移到右边：

x² + 6x = 7

Take half of 6, which is 3, and square it to get 9. Add 9 to both sides:

取 6 的一半，即 3，平方得到 9。两边同时加 9：

x² + 6x + 9 = 16

The left side is now a perfect square:

左边现在是一个完全平方式：

(x + 3)² = 16

Take the square root of both sides:

两边开平方根：

x + 3 = ±4

So x = 1 or x = −7.

所以 x = 1 或 x = −7。

For a quadratic with a coefficient of x² not equal to 1, divide the whole equation by that coefficient first, then complete the square.

若 x² 前面的系数不为 1，则先将整个方程除以该系数，然后再配方。

4. The Quadratic Formula | 二次求根公式

The quadratic formula is the most general method. For any equation ax² + bx + c = 0, the solutions are given by:

二次求根公式是最通用的方法。对于任意方程 ax² + bx + c = 0，解由下式给出：

x = (−b ± √(b² − 4ac)) / (2a)

This formula is derived from completing the square. It works for every quadratic equation, including those with irrational or complex roots.

这个公式由配方法推导而来。它适用于所有二次方程，包括无理根或复数根的情况。

Example 3: Solve 2x² − 4x − 3 = 0 using the formula.

例 3：用公式法解方程 2x² − 4x − 3 = 0。

Here a = 2, b = −4, and c = −3. Substitute into the formula:

这里 a = 2，b = −4，c = −3。代入公式：

x = (4 ± √((−4)² − 4×2×(−3))) / (2×2)

Calculate the discriminant:

计算判别式：

b² − 4ac = 16 + 24 = 40

So:

所以：

x = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± (√10)/2

Therefore the two solutions are x = 1 + (√10)/2 and x = 1 − (√10)/2.

因此两个解为 x = 1 + (√10)/2 和 x = 1 − (√10)/2。

5. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant. It tells us the nature of the roots without fully solving the equation.

根号内的表达式 b² − 4ac 称为判别式。它无需完整解方程就能告诉我们根的性质。

If b² − 4ac > 0, there are two distinct real roots.

若 b² − 4ac > 0，方程有两个不同的实数根。
If b² − 4ac = 0, there is exactly one repeated real root.

若 b² − 4ac = 0，方程有一个重根（两个相等的实数根）。
If b² − 4ac < 0, there are no real roots; the solutions are complex.

若 b² − 4ac < 0，方程没有实数根；解为复数。

Example 4: Determine the nature of the roots of 3x² + 2x + 5 = 0.

例 4：判断方程 3x² + 2x + 5 = 0 的根的性质。

Here a = 3, b = 2, c = 5. The discriminant is 2² − 4×3×5 = 4 − 60 = −56. Since the discriminant is negative, the equation has no real roots.

这里 a = 3，b = 2，c = 5。判别式为 2² − 4×3×5 = 4 − 60 = −56。因为判别式为负数，所以方程没有实数根。

The discriminant also helps in geometry: a tangent to a curve corresponds to a repeated root, so the discriminant equals zero.

判别式在几何中也有应用：切线与曲线对应重根，此时判别式等于零。

6. Solving by Using the Graph | 图像法

Quadratic equations can also be solved graphically. The roots of ax² + bx + c = 0 are the x-coordinates where the curve y = ax² + bx + c crosses the x-axis.

二次方程也可以通过图像求解。方程 ax² + bx + c = 0 的根，就是曲线 y = ax² + bx + c 与 x 轴交点的横坐标。

If the curve intersects the x-axis at two points, there are two real roots. If it touches the axis at one point, there is one repeated root. If it never meets the x-axis, there are no real roots.

如果曲线与 x 轴有两个交点，则有两个实数根。如果曲线与 x 轴相切于一点，则有一个重根。如果曲线始终不与 x 轴相交，则没有实数根。

Graphical methods are useful for approximating roots, especially when the equation cannot be factorised easily. However, they are less accurate than algebraic methods unless you solve the equation exactly.

图像法适合估算根，尤其当方程不易因式分解时。但除非你精确求解，否则图像法不如代数方法精确。

To draw a quadratic graph, choose a range of x-values, calculate corresponding y-values, plot the points, and join them with a smooth curve. The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upwards; if a < 0, it opens downwards.

要画二次函数图像，可选取一系列 x 值，计算对应的 y 值，描点并用平滑曲线连接。y = ax² + bx + c 的图像是抛物线。若 a > 0，开口向上；若 a < 0，开口向下。

7. Rearranging Equations into Standard Form | 化为标准形式

Not every quadratic equation is given in standard form. Often you must first expand, simplify, or rearrange the equation.

并非所有二次方程都以标准形式给出。通常你需要先展开、化简或重新整理方程。

Example 5: Solve (x − 3)(x + 2) = 14.

例 5：解方程 (x − 3)(x + 2) = 14。

First expand the left side:

先展开左边：

x² − x − 6 = 14

Move 14 to the left:

把 14 移到左边：

x² − x − 20 = 0

Factorise:

因式分解：

(x − 5)(x + 4) = 0

So x = 5 or x = −4.

所以 x = 5 或 x = −4。

Always expand and collect all terms on one side before applying a solution method. Never try to solve by taking square roots when the other side is not zero.

在应用任何解法之前，务必展开并整理所有项到等号一边。切勿在另一边不为零时直接开平方。

8. Word Problems Involving Quadratics | 二次方程应用题

Many real-world problems lead to quadratic equations. The first step is always to define a variable, then form the equation from the given information.

许多实际问题会归结为二次方程。第一步总是先设未知数，然后根据已知信息建立方程。

Example 6: The area of a rectangle is 36 cm², and its length is 5 cm more than its width. Find the width.

例 6：一个矩形的面积为 36 cm²，其长比宽多 5 cm。求宽。

Let the width be x cm. Then the length is (x + 5) cm. The area is x(x + 5) = 36, so:

设宽为 x cm，则长为 (x + 5) cm。面积为 x(x + 5) = 36，因此：

x² + 5x − 36 = 0

Factorise: (x + 9)(x − 4) = 0, so x = −9 or x = 4. Since width cannot be negative, the width is 4 cm.

因式分解：(x + 9)(x − 4) = 0，所以 x = −9 或 x = 4。宽度不能为负，因此宽为 4 cm。

When solving word problems, always interpret the solutions in the context and discard any that do not make sense.

解应用题时，务必结合题意解释每个解，并舍弃不合理的结果。

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

For a quadratic equation ax² + bx + c = 0, if the roots are α and β, then:

对于二次方程 ax² + bx + c = 0，若根为 α 和 β，则：

α + β = −b/a

αβ = c/a

These relationships are useful for checking answers or constructing equations with given roots. For example, if the roots are 3 and −2, then the equation is x² − (3 + (−2))x + (3×(−2)) = 0, which is x² − x − 6 = 0.

这些关系可用于检验答案或根据已知根构造方程。例如，若根为 3 和 −2，则方程为 x² − (3 + (−2))x + (3×(−2)) = 0，即 x² − x − 6 = 0。

Remember that this applies only when the quadratic is in the form ax² + bx + c = 0, with all terms on one side.

请注意，仅当二次方程化为 ax² + bx + c = 0 的形式（所有项都在一侧）时才适用。

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

Several common mistakes can cost marks in exams. Being aware of them helps you avoid them.

一些常见错误会在考试中失分。了解它们有助于避免这些错误。

Forgetting to rearrange the equation to zero before factorising or using the formula. Always write ax² + bx + c = 0.

忘记先把方程整理成等于零的形式再因式分解或套公式。务必写成 ax² + bx + c = 0。
Taking the square root only of some terms instead of both whole sides. For example, from x² = 9, x = ±3, not just x = 3.

只对部分项开平方而不是对等式两边整体开平方。例如由 x² = 9，得 x = ±3，而不是仅 x = 3。
Mixing up the sign of b in the quadratic formula. Always write x = (−b ± √…) / (2a), paying attention to negative values.

在二次求根公式中弄错 b 的符号。始终写成 x = (−b ± √…) / (2a)，注意负数的情况。
Forgetting that factorisation must give the original expression when expanded. Check your factors.

忘记因式分解后展开必须等于原表达式。请检查你的因式是否正确。
Dropping negative solutions in word problems without justification.

在应用题中未加说明直接舍弃负数解。

To avoid careless errors, substitute your roots back into the original equation to verify them.

为避免粗心错误，可将解代回原方程进行验算。

11. Choosing the Best Method | 选择最佳解法

In an exam, you should choose the most efficient method for each equation. The table below summarises the key indications.

在考试中，你应该为每个方程选择最高效的解法。下表总结了主要判断依据。

Method	When to Use	Example
Factorisation	When the expression clearly factors into two binomials	x² + 5x + 6 = 0
Completing the square	When the coefficient of x² is 1 and b is even, or when the roots involve surds	x² − 4x + 1 = 0
Quadratic formula	When factorisation is not obvious, or the equation has irrational roots	2x² − 7x + 3 = 0
Graphical method	When you are asked to estimate roots from a graph	Sketch y = x² − x − 2

If a question states “solve by factorising”, you must use that method even if another method would work. Otherwise, choose the fastest reliable method.

如果题目要求“用因式分解法求解”，即使其他方法可行也必须使用该方法。否则，选择最快且可靠的方法。

12. Exam Practice and Conclusion | 练习与总结

To master quadratic equations, you need consistent practice. Work through past-paper questions, focusing on accuracy and speed. Always show clear steps, as method marks are awarded even if your final answer has a small numerical error.

要掌握二次方程，你需要持续练习。认真做历年真题，注重准确性和速度。务必写出清晰步骤，因为即使最终答案有小数值错误，步骤分仍然可以获得。

Summarising the key points: a quadratic equation has the form ax² + bx + c = 0. You can solve it by factorisation, completing the square, the quadratic formula, or a graph. Use the discriminant b² − 4ac to determine the nature of the roots. For word problems, define the variable and interpret your answers logically.

总结要点：二次方程形式为 ax² + bx + c = 0。你可以通过因式分解、配方、求根公式或图像来求解。利用判别式 b² − 4ac 判断根的性质。对于应用题，先设未知数并合理解释答案。

With regular revision and careful practice, solving quadratic equations will become a reliable skill in your IGCSE mathematics toolkit.

通过规律复习和认真练习，解二次方程将成为你 IGCSE 数学考试中的可靠技能。

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Solving Quadratic Equations | 解二次方程