Solving Quadratic Equations | 二次方程解法解析

📚 Solving Quadratic Equations | 二次方程解法解析

Quadratics are one of the most important topics in IGCSE Mathematics. This article covers everything you need for the exam: expanding, factorising, solving, sketching, and applying quadratic equations.

二次方程是 IGCSE 数学中最重要的话题之一。本文将涵盖考试所需的一切：展开、因式分解、求解、画图以及实际应用。

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

A quadratic expression has the general form ax² + bx + c, where a, b, and c are constants and a ≠ 0. When we set it equal to zero, we get a quadratic equation.

二次表达式的一般形式为 ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。当令其等于零时，便得到二次方程。

The highest power of x is 2, giving the graph a parabolic shape. The coefficient a controls whether the parabola opens upward (a > 0) or downward (a < 0).

x 的最高次数是 2，使图像呈抛物线形状。系数 a 决定抛物线开口朝上（a > 0）还是朝下（a < 0）。

Standard form: ax² + bx + c = 0 标准形式：ax² + bx + c = 0
Roots are the x-values where the graph crosses the x-axis. 根是图像与 x 轴交点的 x 值。

2. Expanding Double Brackets | 展开双括号

To expand (mx + p)(nx + q), multiply each term in the first bracket by each term in the second bracket. Use FOIL: First, Outer, Inner, Last.

展开 (mx + p)(nx + q) 时，用第一个括号中的每一项乘以第二个括号中的每一项。可运用 FOIL 法则：First（首）、Outer（外）、Inner（内）、Last（尾）。

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

Simplify by collecting like terms. This process is essential for understanding factorisation in reverse.

合并同类项进行化简。这一过程对于反向理解因式分解至关重要。

3. Factorising Quadratics with Leading Coefficient 1 | 二次项系数为 1 的因式分解

When a = 1, we look for two numbers that multiply to give c and add to give b. The factorised form is (x + m)(x + n).

当 a = 1 时，我们需要找到两个数，它们相乘得 c，相加得 b。因式分解形式为 (x + m)(x + n)。

For example, factorise x² + 7x + 12. Find two numbers with product 12 and sum 7: 3 and 4.

例如，分解 x² + 7x + 12。找到乘积为 12、和为 7 的两个数：3 和 4。

x² + 7x + 12 = (x + 3)(x + 4)

Always check the signs. If c is positive, both factors have the same sign as b. If c is negative, the factors have opposite signs.

务必注意符号。若 c 为正，则两个因子的符号与 b 相同；若 c 为负，则两个因子符号相反。

4. Factorising Quadratics with Leading Coefficient ≠ 1 | 二次项系数不为 1 的因式分解

When a is not 1, we can use the method of splitting the middle term. Multiply a and c, then find two numbers that multiply to ac and add to b.

当 a 不为 1 时，可使用“拆中项”的方法。先计算 a × c，再找两个数，使其乘积为 ac、和为 b。

For example, factorise 2x² + 7x + 3. Here a × c = 6, and b = 7. Two numbers are 6 and 1.

例如，分解 2x² + 7x + 3。这里 a × c = 6，b = 7。两个数为 6 和 1。

2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)

Alternatively, use the “box method” or trial and error. Practise each method until you find one that feels reliable.

也可以使用“表格法”或试错法。多练习每种方法，直到找到一种你认为可靠的方法。

5. Difference of Two Squares | 平方差公式

The special pattern x² – y² = (x + y)(x – y) is called the difference of two squares. It appears frequently in IGCSE questions.

特殊形式 x² – y² = (x + y)(x – y) 称为平方差公式。这在 IGCSE 考题中经常出现。

Examples include: x² – 9 = (x + 3)(x – 3), and 4x² – 25 = (2x + 5)(2x – 5).

例如：x² – 9 = (x + 3)(x – 3)，以及 4x² – 25 = (2x + 5)(2x – 5)。

Recognising this pattern allows you to factorise quickly and also helps when solving equations.

识别这一模式可以快速进行因式分解，也有助于解方程。

6. Solving Quadratic Equations by Factorisation | 用因式分解法解二次方程

If a product of two factors equals zero, then at least one factor must be zero. This is the zero product property.

若两个因子的乘积为零，则至少有一个因子为零。这就是零乘积性质。

To solve x² – 5x + 6 = 0, factorise first: (x – 2)(x – 3) = 0.

解 x² – 5x + 6 = 0，先因式分解： (x – 2)(x – 3) = 0。

x – 2 = 0 → x = 2; x – 3 = 0 → x = 3

Always rearrange the equation to the form ax² + bx + c = 0 before factorising. Do not divide by a variable factor, as this may lose a root.

在因式分解前，务必先将方程整理为 ax² + bx + c = 0 的形式。不要除以含变量的因子，否则可能会丢根。

7. Completing the Square | 配方法

Completing the square rewrites a quadratic as a(x + p)² + q. This is useful for finding the vertex of a parabola and solving equations.

配方法将二次式改写为 a(x + p)² + q。这有助于求抛物线的顶点以及解方程。

For x² + 6x + 2, take half of 6 (which is 3), square it (9), then adjust:

对于 x² + 6x + 2，取 6 的一半（即 3），平方得 9，然后调整：

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

To solve, set the expression equal to zero and isolate the square. Remember to take both positive and negative square roots.

解方程时，令表达式等于零，再将平方项单独放在一边。记住取正负两个平方根。

8. The Quadratic Formula | 求根公式

The quadratic formula can solve any quadratic equation ax² + bx + c = 0. It is especially useful when factorisation is difficult or impossible.

求根公式可以解任意二次方程 ax² + bx + c = 0。当因式分解困难或无法进行时，它尤其有用。

x = (−b ± √(b² − 4ac)) / (2a)

Substitute the values of a, b, and c carefully. Write down both solutions, and use a calculator only for the final decimal approximation.

仔细代入 a、b、c 的值。写出两个解，仅在最后近似小数时使用计算器。

For example, for 2x² + 3x − 2 = 0, a = 2, b = 3, c = −2.

例如，对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = −2。

x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4 → x = 0.5 or x = −2

9. The Discriminant | 判别式

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

b² − 4ac 称为判别式。它无需解方程即可告诉我们根的性质。

Discriminant (Δ)	Nature of Roots
Δ > 0	Two real distinct roots
Δ = 0	One real repeated root
Δ < 0	No real roots (complex roots)

If the discriminant is a perfect square, the equation factorises and the roots are rational. Otherwise, the roots are irrational.

若判别式是完全平方数，则方程可因式分解且根为有理数；否则根为无理数。

10. Sketching Quadratic Graphs | 画二次函数图像

To sketch y = ax² + bx + c, identify three key features: the roots, the y-intercept, and the vertex.

画 y = ax² + bx + c 的草图时，需确定三个关键要素：根、y 轴截距和顶点。

Roots occur where y = 0 (solve the equation). 根出现在 y = 0 处（解方程）。
The y-intercept is c. y 轴截距是 c。
The vertex is found by completing the square or using x = −b/(2a). 顶点可通过配方法或 x = −b/(2a) 求得。

Vertex x-coordinate = −b/(2a); then substitute to find y.

若没有实根，则图像完全位于 x 轴上方（a > 0）或下方（a < 0）。

11. Applications and Problem Solving | 实际应用与解题

Quadratic equations appear in geometry, physics, and business problems. Common examples include areas of rectangles, projectile motion, and profit maximisation.

二次方程出现在几何、物理和商业问题中。常见例子包括矩形面积、抛体运动和利润最大化。

For instance, a rectangle has length (x + 4) and width (x − 1), with area 30. Then (x + 4)(x − 1) = 30.

例如，一个矩形的长为 (x + 4)，宽为 (x − 1)，面积为 30。则 (x + 4)(x − 1) = 30。

x² + 3x − 4 = 30 → x² + 3x − 34 = 0

Solve using the quadratic formula, and always check that the answer makes sense in the context (negative lengths are impossible).

使用求根公式求解，并始终检查答案在具体情境中是否合理（长度不可能为负）。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students lose marks on quadratics because of small sign errors or forgetting to rearrange the equation. Below are the most important exam tips.

许多学生因符号错误或忘记整理方程而在二次方程上丢分。以下是最重要的考试技巧。

Always write the equation in standard form first. 先写出标准形式的方程。
Check factorisation by expanding your answer. 通过展开检查因式分解是否正确。
Use the discriminant to predict the number of roots. 用判别式预判根的个数。
When using the formula, double-check a, b, and c. 使用公式时，反复检查 a、b、c 的值。
Give exact roots unless rounding is asked. 除非题目要求近似值，否则给出精确根。
Do not confuse (x + 3)² with x² + 9. 不要混淆 (x + 3)² 与 x² + 9。

Practise many past-paper questions. Quadratics are a foundation for later topics such as inequalities and functions.

多做历年真题。二次方程是后续不等式和函数等主题的基础。

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Solving Quadratic Equations | 二次方程解法解析