📚 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
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
(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
x + 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
n 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
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。
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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