Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, geometry, and even trigonometry problems. Mastering the different methods of solving quadratics will not only earn you marks in Paper 2 and Paper 4, but also build a strong foundation for A-Level mathematics.

二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图象、几何甚至三角学问题中。掌握解二次方程的不同方法，不仅能在 Paper 2 和 Paper 4 中为你赢得分数，还能为 A-Level 数学打下坚实的基础。

1. What Is a Quadratic Equation? | 什么是一元二次方程

A quadratic equation is an equation where the highest power of the unknown variable is 2. The word ‘quadratic’ comes from the Latin word ‘quadratus’, which means square. In its simplest form, the variable appears as x², and the equation must be rearranged so that one side equals zero.

二次方程是未知数的最高次数为 2 的方程。英文单词 ‘quadratic’ 源自拉丁语 ‘quadratus’，意为”平方”。在最基本的形式中，未知数表现为 x²，方程必须经过整理，使一边等于零。

The general form of a quadratic equation is written as follows:

ax² + bx + c = 0, where a ≠ 0

Here a, b, and c are constants, and a cannot be zero. If a were zero, the equation would become linear, not quadratic. In IGCSE exams, you must be able to identify these coefficients quickly and accurately.

其中 a、b、c 是常数，且 a 不能为零。如果 a 为零，方程就变成了一次方程，而不是二次方程。在 IGCSE 考试中，你必须能够快速准确地识别这些系数。

For example, in the equation 2x² − 3x + 1 = 0, the coefficients are a = 2, b = −3, and c = 1. Recognising the signs of negative coefficients is a common source of errors, so always read the equation carefully.

例如，在方程 2x² − 3x + 1 = 0 中，系数分别为 a = 2，b = −3，c = 1。识别负系数的符号是常见的错误来源，因此务必仔细阅读方程。

2. Three Forms of Quadratic Equations | 二次方程的三种形式

A quadratic equation can be written in several equivalent forms, and each form is useful for a different purpose. Understanding all three forms allows you to choose the most efficient solving method in an exam.

二次方程可以用几种等价的形式书写，每种形式都有不同的用途。理解这三种形式，可以让你在考试中选择最高效的求解方法。

Form / 形式	Expression / 表达式	Main Use / 主要用途
General / 一般式	ax² + bx + c = 0	Identify coefficients / 识别系数
Factorised / 因式分解式	(px + q)(rx + s) = 0	Read roots directly / 直接读出根
Completed square / 配方式	a(x + h)² + k = 0	Find turning point / 求顶点坐标

The general form is the starting point for all methods. The factorised form is the quickest way to find solutions, while the completed square form directly reveals the vertex of the parabola.

一般式是所有方法的起点。因式分解式是求根最快的方法，而配方式则能直接揭示抛物线的顶点。

3. Solving by Factorisation | 因式分解法

Factorisation is the most elegant method and is often the fastest if the quadratic has simple integer roots. The method relies on the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

因式分解法是最优雅的方法，如果二次方程具有简单的整数根，它通常也是最快的方法。该方法依赖于零乘积性质：如果两个表达式的乘积为零，那么至少其中一个必须为零。

Follow these four steps to solve any quadratic by factorisation:

Step 1: Rearrange the equation so that all terms are on one side, giving ax² + bx + c = 0.

第一步：整理方程，将所有项移到一边，得到 ax² + bx + c = 0。
Step 2: Factorise the quadratic expression into two brackets.

第二步：将二次表达式因式分解为两个括号。
Step 3: Apply the zero product property by setting each bracket equal to zero.

第三步：应用零乘积性质，令每个括号等于零。
Step 4: Solve the two linear equations to find the two roots.

第四步：解两个一次方程，求出两个根。

Worked example: solve x² − 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give −5, which are −2 and −3.

例题：解 x² − 5x + 6 = 0。我们需要找两个数，它们相乘得 6，相加得 −5，即 −2 和 −3。

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0

x − 2 = 0 or x − 3 = 0 → x = 2 or x = 3

Always check your answers by substituting each root back into the original equation. A quick mental check can catch sign errors and arithmetic mistakes.

务必通过将每个根代回原方程来检查答案。快速的心算检查可以捕捉符号错误和算术错误。

4. Solving by the Quadratic Formula | 公式法

The quadratic formula is a universal method that works for every quadratic equation, even when factorisation is impossible or difficult. It is given on the IGCSE formula sheet, but you must know exactly how to substitute values correctly.

求根公式是一种通用方法，适用于所有二次方程，即使因式分解无法进行或很困难。IGCSE 公式表中会给出该公式，但你必须知道如何正确代入数值。

For a quadratic equation ax² + bx + c = 0, the solutions are given by:

x = (−b ± √(b² − 4ac)) / (2a)

The symbol ± means that you calculate two values: one with addition and one with subtraction. This is why a quadratic equation always has two roots, two equal roots, or no real roots.

符号 ± 表示你需要计算两个值：一个用加法，一个用减法。这就是为什么二次方程总是有两个根、两个相等的根，或没有实数根。

Worked example: solve 2x² + 7x − 4 = 0. Here a = 2, b = 7, and c = −4. Substituting into the formula gives:

例题：解 2x² + 7x − 4 = 0。这里 a = 2，b = 7，c = −4。代入公式得：

x = (−7 ± √(7² − 4 × 2 × (−4))) / (2 × 2) = (−7 ± √(49 + 32)) / 4

x = (−7 ± 9) / 4 → x = 0.5 or x = −4

Notice that when c is negative, the term 4ac becomes positive after multiplication. A common mistake is to write −4 × 2 × (−4) incorrectly, so always write out the substitution fully before calculating.

注意，当 c 为负数时，4ac 相乘后变为正数。一个常见错误是写错 −4 × 2 × (−4)，因此务必在计算前完整写出代入过程。

5. Solving by Completing the Square | 配方法

Completing the square transforms a quadratic into the form a(x + h)² + k = 0. This method is slower than factorisation, but it is extremely useful for finding the turning point of a graph and for solving equations that cannot be factorised.

配方法将二次方程转化为 a(x + h)² + k = 0 的形式。这种方法比因式分解慢，但对于求图象的顶点以及解无法因式分解的方程非常有用。

The method works as follows. Take the coefficient of x, halve it, and square it. Then add and subtract this value inside the expression.

该方法步骤如下：取 x 的系数，将其减半，然后平方。接着在表达式中加上并减去这个值。

Worked example: solve x² + 6x − 3 = 0 by completing the square. The coefficient of x is 6, so half of it is 3, and 3² = 9.

例题：用配方法解 x² + 6x − 3 = 0。x 的系数是 6，所以一半是 3，3² = 9。

x² + 6x − 3 = 0 → (x + 3)² − 9 − 3 = 0

(x + 3)² − 12 = 0 → (x + 3)² = 12

x + 3 = ±√12 → x = −3 ± 2√3

This gives the exact answers x = −3 + 2√3 and x = −3 − 2√3. Note that √12 simplifies to 2√3, so always simplify surds in your final answer unless the question asks for a decimal.

这样得到精确答案 x = −3 + 2√3 和 x = −3 − 2√3。注意 √12 可化简为 2√3，因此除非题目要求小数，否则最终答案中应化简根式。

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

The discriminant is the part of the quadratic formula under the square root sign, b² − 4ac. It tells us the nature of the roots without actually solving the equation. This is a frequently tested concept in IGCSE non-calculator papers.

判别式是求根公式中根号下的部分，即 b² − 4ac。它无需实际解方程就能告诉我们根的性质。这是 IGCSE 非计算器试卷中经常考查的概念。

Discriminant Δ / 判别式 Δ	Nature of Roots / 根的性质	Graph Meaning / 图象含义
b² − 4ac > 0	Two distinct real roots / 两个不相等的实数根	Graph cuts x-axis at two points / 图象与 x 轴交于两点
b² − 4ac = 0	One repeated real root / 两个相等的实数根	Graph touches x-axis at one point / 图象与 x 轴相切于一点
b² − 4ac < 0	No real roots / 没有实数根	Graph does not meet x-axis / 图象不与 x 轴相交

For example, consider x² − 4x + 5 = 0. The discriminant is b² − 4ac = (−4)² − 4 × 1 × 5 = 16 − 20 = −4. Since −4 < 0, this equation has no real roots, and its graph lies entirely above the x-axis.

例如，考虑 x² − 4x + 5 = 0。判别式为 b² − 4ac = (−4)² − 4 × 1 × 5 = 16 − 20 = −4。由于 −4 < 0，此方程没有实数根，其图象完全位于 x 轴上方。

7. Sketching Quadratic Graphs | 二次函数图象

The graph of a quadratic function y = ax² + bx + c is a parabola. Its shape depends on the sign of a: if a > 0, the parabola opens upwards like a U; if a < 0, it opens downwards like an upside-down U.

二次函数 y = ax² + bx + c 的图象是抛物线。其开口方向取决于 a 的符号：若 a > 0，抛物线开口向上，形如 U；若 a < 0，抛物线开口向下，形如倒过来的 U。

To sketch a quadratic graph accurately, you need three key features: the roots, the y-intercept, and the turning point. The roots are found by solving the quadratic equation, the y-intercept is simply the value of c, and the turning point can be found by completing the square.

要准确画出二次函数图象，你需要三个关键特征：根、y 轴截距和顶点。根通过解二次方程求得，y 轴截距就是 c 的值，而顶点可以通过配方法求得。

Worked example: sketch y = x² − 4x + 3. First, factorise to find the roots: (x − 1)(x − 3) = 0, so x = 1 and x = 3. The y-intercept is 3. Completing the square gives y = (x − 2)² − 1, so the turning point is (2, −1).

例题：画出 y = x² − 4x + 3 的图象。首先因式分解求根：(x − 1)(x − 3) = 0，所以 x = 1 和 x = 3。y 轴截距为 3。配方得 y = (x − 2)² − 1，因此顶点为 (2, −1)。

In an exam, label the x-intercepts, the y-intercept, and the vertex clearly. Since a = 1 > 0, the parabola opens upwards, and the vertex is a minimum point.

在考试中，请清楚标出 x 轴交点、y 轴截距和顶点。由于 a = 1 > 0，抛物线开口向上，顶点是最小值点。

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

For a quadratic equation ax² + bx + c = 0 with roots α and β, there are two important relationships that allow you to check answers or construct equations without solving. These relationships are stated below.

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，有两个重要的关系式，它们可用于检查答案或在不解方程的情况下构造方程。这些关系如下所示。

Sum of roots: α + β = −b/a

Product of roots: α × β = c/a

Consider the equation x² + 5x + 6 = 0. The roots are x = −2 and x = −3. Their sum is −5, which equals −b/a = −5/1 = −5. Their product is 6, which equals c/a = 6/1 = 6. The relationships hold perfectly.

考虑方程 x² + 5x + 6 = 0。根为 x = −2 和 x = −3。它们的和为 −5，等于 −b/a

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Solving Quadratic Equations | 解一元二次方程