Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most heavily tested topics in the Edexcel IGCSE Mathematics syllabus. They appear in nearly every paper and connect directly to factorisation, graphs, inequalities and word problems. This revision guide breaks down every essential method step by step, with exam-style examples and the common pitfalls that cost students marks.

二次方程是 Edexcel IGCSE 数学考纲中考查频率最高的话题之一。几乎每份试卷都会出现，并直接关联到因式分解、函数图像、不等式和实际应用题。本复习指南将逐步拆解所有核心方法，配以考试风格的例题，并指出学生容易失分的常见陷阱。

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

A quadratic equation is a polynomial equation in which the highest power of the unknown variable is 2. The general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. For example, 2x² − 3x + 1 = 0 is quadratic, while 2x − 1 = 0 is linear and x³ − 4x = 0 is cubic.

二次方程是未知数最高次数为 2 的多项式方程。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。例如，2x² − 3x + 1 = 0 是二次方程，而 2x − 1 = 0 是一次方程，x³ − 4x = 0 是三次方程。

Key features you must know:

你必须掌握的关键特征如下：

The coefficient a controls the shape of the graph. 系数 a 决定图像的开口方向。
The equation may have two, one or zero real solutions. 方程可能有两个、一个或零个实数解。
You can always check your answer by substitution. 你始终可以把答案代回原式进行检验。

2. Solving by Factorisation | 用因式分解法求解

Factorisation is the fastest method when the quadratic has simple integer factors. First rearrange the equation so that one side is zero. Then write the quadratic as a product of two brackets, and solve each bracket equal to zero because if the product is zero, at least one factor must be zero.

当二次式含有简单的整数因子时，因式分解是最快捷的方法。首先整理方程，使一边为零；然后把二次式写成两个括号相乘；由于乘积为零时至少有一个因子为零，所以令每个括号分别等于零并求解。

Follow these steps:

按以下步骤操作：

Rearrange to the form ax² + bx + c = 0. 整理成 ax² + bx + c = 0 的形式。
Factorise the left-hand side into two brackets. 把左边因式分解为两个括号。
Set each factor equal to zero. 令每个因子等于零。
Solve the two linear equations. 解这两个一次方程。

Worked example: solve x² − 5x + 6 = 0. The factors of +6 that add to −5 are −2 and −3, so (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.

例题：解 x² − 5x + 6 = 0。乘积为 +6 且和为 −5 的两个数是 −2 和 −3，因此 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。

(x − 2)(x − 3) = 0 → x = 2 or x = 3

3. Solving by Completing the Square | 用配方法求解

Completing the square rewrites x² + bx as a perfect square minus a constant. This method works for every quadratic, including those that do not factorise nicely. It is also essential for finding the maximum or minimum point of a quadratic graph.

配方法把 x² + bx 改写成一个完全平方减去一个常数。这种方法适用于所有二次方程，包括不易因式分解的情况。它也是求二次函数图像最大值或最小值点的关键工具。

Worked example: solve x² + 6x + 2 = 0 by completing the square.

例题：用配方法解 x² + 6x + 2 = 0。

Half of 6 is 3, so (x + 3)² = x² + 6x + 9. We have +2, so subtract 7 to balance: (x + 3)² − 7 = 0.

6 的一半是 3，所以 (x + 3)² = x² + 6x + 9。原式常数项为 +2，因此减去 7 保持平衡：(x + 3)² − 7 = 0。

(x + 3)² = 7 → x + 3 = ±√7 → x = −3 ± √7

Therefore the two solutions are x = −3 + √7 and x = −3 − √7. Leave your answer in surd form unless the question asks for a decimal.

因此两个解为 x = −3 + √7 和 x = −3 − √7。除非题目要求小数，否则应保留根式形式。

4. The Quadratic Formula | 二次求根公式

The quadratic formula solves any quadratic equation of the form ax² + bx + c = 0. You should memorise it because it always works, even when factorisation fails or completing the square is too slow.

二次求根公式可以解任何形如 ax² + bx + c = 0 的二次方程。你必须牢记这个公式，因为它始终有效，即使因式分解失败或配方法太慢也能使用。

x = (−b ± √(b² − 4ac)) / 2a

Worked example: solve 2x² + 3x − 2 = 0 using the formula. Here a = 2, b = 3, c = −2.

例题：用公式法解 2x² + 3x − 2 = 0。此时 a = 2，b = 3，c = −2。

x = (−3 ± √(9 − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √25) / 4

So x = (−3 + 5) / 4 = 0.5, or x = (−3 − 5) / 4 = −2. Always substitute back to check that both values satisfy the original equation.

因此 x = (−3 + 5) / 4 = 0.5，或 x = (−3 − 5) / 4 = −2。务必代回原式检验两个值是否都成立。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root, Δ = b² − 4ac. It tells you how many real roots the equation has without fully solving it.

判别式是二次求根公式中根号内的部分，即 Δ = b² − 4ac。它可以在不完全求解的情况下告诉我们方程有多少个实数根。

Discriminant 判别式	Nature of roots 根的情况
b² − 4ac > 0	Two distinct real roots 两个不同的实数根
b² − 4ac = 0	One repeated real root 一个相等的实数根
b² − 4ac < 0	No real roots 没有实数根

If b² − 4ac is a perfect square and a, b, c are rational, the equation can be solved by factorisation. If the discriminant is negative, the graph never touches the x-axis.

当 a、b、c 为有理数且 b² − 4ac 是一个完全平方数时，方程可以用因式分解求解。当判别式为负数时，图像与 x 轴没有交点。

6. Drawing and Interpreting Graphs | 二次函数图像的绘制与解读

The graph of y = ax² + bx + c is always a curve called a parabola. Its shape depends on the sign of a: if a > 0 the curve is U-shaped and has a minimum point; if a < 0 the curve is n-shaped and has a maximum point.

y = ax² + bx + c 的图像总是一条称为抛物线的曲线。其开口方向取决于 a 的符号：当 a > 0 时，曲线呈 U 形并具有最小值点；当 a < 0 时，曲线呈 n 形并具有最大值点。

To sketch a quadratic graph, identify these features:

画二次函数草图时需要确定以下特征：

The y-intercept is c, where the curve crosses the y-axis. y 轴截距为 c，即曲线与 y 轴的交点。
The roots are the solutions of ax² + bx + c = 0. 根是方程 ax² + bx + c = 0 的解。
The vertex is the turning point (maximum or minimum). 顶点是转折点（最大值或最小值）。
The axis of symmetry is a vertical line through the vertex. 对称轴是经过顶点的一条竖直线。

Always plot at least five points: the vertex, the two roots, and the y-intercept together with its mirror point.

画图时至少要标出五个点：顶点、两个根、y 轴截距及其对称点。

7. Finding the Vertex and Line of Symmetry | 求顶点与对称轴

The x-coordinate of the vertex is always x = −b / (2a). Substitute this value into the equation to find the y-coordinate. The line of symmetry is simply the vertical line x = −b / (2a).

顶点的 x 坐标始终为 x = −b / (2a)。把这个值代入原方程即可求出 y 坐标。对称轴就是竖直线 x = −b / (2a)。

Worked example: find the vertex of y = x² − 4x + 1.

例题：求 y = x² − 4x + 1 的顶点。

Here a = 1 and b = −4, so x = −(−4) / (2 × 1) = 2. Then y = 2² − 4 × 2 + 1 = −3. The vertex is (2, −3) and its axis of symmetry is x = 2.

这里 a = 1，b = −4，所以 x = −(−4) / (2 × 1) = 2。于是 y = 2² − 4 × 2 + 1 = −3。顶点为 (2, −3)，对称轴为 x = 2。

Alternatively, complete the square: y = (x − 2)² − 3, so the vertex is (−p, q) = (2, −3).

另一种方法是用配方法：y = (x − 2)² − 3，因此顶点为 (−p, q) = (2, −3)。

8. Quadratic Inequalities | 二次不等式

Solving a quadratic inequality requires a graph sketch. First solve the corresponding equation to find the critical values, then test which regions satisfy the inequality sign.

解二次不等式需要借助函数草图。首先解对应的方程找到临界值，然后判断哪个区间满足不等号的方向。

Worked example: solve x² − x − 6 < 0.

例题：解 x² − x − 6 < 0。

Factorise: (x − 3)(x + 2) = 0, so the critical values are x = 3 and x = −2. Sketching the U-shaped graph shows the curve is below the x-axis between these two values.

因式分解得 (x − 3)(x + 2) = 0，因此临界值为 x = 3 和 x = −2。画出 U 形草图可以发现，曲线在这两个值之间的部分位于 x 轴下方。

−2 < x < 3

Remember: for ‘less than zero’ the solution is the interval between the roots; for ‘greater than zero’ the solution is two separate outer regions.

记住：对于“小于零”，解在两个根之间；对于“大于零”，解是两个外侧区间的并集。

9. Problem Solving with Quadratics | 二次方程应用题

Many IGCSE questions place quadratics in real-world contexts, especially geometry and motion. You must translate the words into an equation, solve it, and then reject any answer that makes no physical sense.

许多 IGCSE 题目将二次函数置于实际情境中，尤其是几何与运动问题。你必须把文字转化为方程，求解后舍去不符合实际意义的答案。

Worked example: a rectangle has length (x + 3) cm and width (x − 1) cm. Its area is 21 cm². Find x.

例题：一个长方形的长为 (x + 3) cm，宽为 (x − 1) cm，面积为 21 cm²。求 x。

(x + 3)(x − 1) = 21 → x² + 2x − 3 = 21 → x² + 2x − 24 = 0

Factorise: (x + 6)(x − 4) = 0, so x = −6 or x = 4. A length cannot be negative, so x = 4 and the original dimensions are 7 cm by 3 cm.

因式分解得 (x + 6)(x − 4) = 0，所以 x = −6 或 x = 4。长度不可能为负数，因此 x = 4，原长方形的尺寸为 7 cm × 3 cm。

For projectile motion, use h = ut − 4.9t² when g = 9.8 m/s². Set h = 0 to find when the object hits the ground.

对于抛体运动，当 g = 9.8 m/s² 时使用 h = ut − 4.9t²。令 h = 0 即可求出物体落地的时间。

10. Common Mistakes and Exam Tips | 常见错误与应试技巧

Even strong students lose marks on quadratics through careless errors. Here is a checklist of the most frequent mistakes and how to avoid them.

即使是优秀学生也会因粗心而在二次方程上失分。以下是最高频的错误清单及避免方法。

Forgetting to rearrange the equation to zero before factorising. 因式分解前忘记把方程整理成等于零的形式。
Losing a negative sign when substituting into the quadratic formula. 代入求根公式时漏写负号。
Missing the ± sign when taking a square root. 开平方时漏写 ± 符号。
Rejecting negative roots without justification. 没有说明理由就舍去负根。
Confusing the x-coordinate of the vertex: it is −b / (2a), not b / (2a). 混淆顶点 x 坐标：应为 −b / (2a)，而不是 b / (2a)。
Writing x = 2 and x = 2 as two roots when the discriminant is zero. 当判别式为零时，把 x = 2 和 x = 2 误写成两个根。

Always show your working clearly; in Edexcel IGCSE, method marks are awarded even if the final answer is wrong. Use a calculator for arithmetic but never for the algebra steps.

务必清晰写出解题过程；在 Edexcel IGCSE 中，即使最终答案错误，只要方法正确仍可获得步骤分。计算器只能用于算术运算，不可代替代数步骤。

11. Quick Revision Summary | 快速复习总结

Use this table to choose the correct method quickly in the exam.

考试时可用此表快速选择正确的方法。

Method 方法	When to use 适用情境
Factorisation 因式分解	Simple integer factors are visible 能直接看出整数因子
Completing the square 配方法	Need the vertex, or the coefficient of x² is 1 需要求顶点，或 x² 的系数为 1
Quadratic formula 求根公式	Factorisation fails or fractions appear 因式分解困难或出现分数
Discriminant 判别式	Only need the number of roots 只需判断根的个数

Master these four tools and quadratics will become one of the most reliable scoring areas in your IGCSE Mathematics paper. Practice past-paper questions until each method feels automatic.

掌握这四种工具后，二次方程将成为你 IGCSE 数学试卷中最稳定的得分点。请反复练习历年真题，直到每种方法都运用自如。

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Mastering Quadratic Equations | 掌握二次方程