Quadratic Equations Mastery | 一元二次方程完全攻略

📚 Quadratic Equations Mastery | 一元二次方程完全攻略

A quadratic equation is one of the most important topics in IGCSE Mathematics (Syllabus 0580/0607). It appears in at least three exam questions on average, including algebra, graphs, and problem-solving. Mastering this topic will strengthen your foundation for calculus and coordinate geometry at A Level.

在 IGCSE 数学（0580/0607 大纲）中，一元二次方程是最核心的考点之一，平均每张试卷至少出现 3 道相关题目，涵盖代数、图像和实际应用题。掌握好这一章，将为 A Level 的微积分和解析几何打下坚实基础。

1. Standard Form of a Quadratic Equation | 一元二次方程的标准形式

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

一元二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。如果 a = 0，方程就变成了一次方程，而非二次方程。

The value of a determines the shape of the graph: if a > 0, the parabola opens upwards (U-shaped); if a < 0, it opens downwards (n-shaped). The constant c gives the y-intercept of the graph.

系数 a 决定抛物线的开口方向：当 a > 0 时，抛物线开口向上（U 形）；当 a < 0 时，开口向下（n 形）。常数 c 对应图像与 y 轴的交点。

Example: In the equation 2x² – 5x + 3 = 0, we identify a = 2, b = -5, c = 3. Always write the equation in descending powers of x before identifying coefficients.

示例：在方程 2x² – 5x + 3 = 0 中，a = 2，b = -5，c = 3。在识别系数之前，务必先将方程按 x 的降幂排列。

Form	Example	a,b,c
ax² + bx + c = 0	3x² + 4x – 7 = 0	a = 3, b = 4, c = -7
x² = 9x – 14	x² – 9x + 14 = 0	a = 1, b = -9, c = 14

2. Expanding Double Brackets | 展开双括号

Expanding (x + p)(x + q) gives x² + (p + q)x + pq. This rule comes from the distributive law: each term in the first bracket multiplies every term in the second bracket.

展开 (x + p)(x + q) 得到 x² + (p + q)x + pq。这个公式源自乘法分配律：第一个括号中的每一项都要与第二个括号中的每一项相乘。

The acronym FOIL (First, Outer, Inner, Last) is a helpful memory aid. For example, (2x + 3)(x – 4) expands as follows:

记忆口诀 FOIL（First 首项、Outer 外项、Inner 内项、Last 末项）非常实用。例如，展开 (2x + 3)(x – 4) 的过程如下：

(2x)(x) + (2x)(-4) + (3)(x) + (3)(-4) = 2x² – 8x + 3x – 12 = 2x² – 5x – 12

Special cases worth memorising: (x + a)² = x² + 2ax + a², and (x – a)² = x² – 2ax + a², and the difference of two squares: (x + a)(x – a) = x² – a².

值得牢记的特殊展开式包括：完全平方公式 (x + a)² = x² + 2ax + a²，(x – a)² = x² – 2ax + a²，以及平方差公式 (x + a)(x – a) = x² – a²。

3. Factorising Quadratics | 因式分解二次式

Factorising is the reverse process of expanding. When a = 1, you look for two numbers whose product is c and whose sum is b. For x² + 7x + 12, we need two numbers multiplying to 12 and adding to 7: those numbers are 3 and 4.

因式分解是展开的逆运算。当 a = 1 时，需要找出两个数，使其乘积为 c、和为 b。对于 x² + 7x + 12，我们需要找乘积为 12、和为 7 的两个数，也就是 3 和 4。

x² + 7x + 12 = (x + 3)(x + 4)

When a ≠ 1, use the method of factors: multiply a and c, then split b into two parts. For 2x² + 7x + 3, multiply 2 × 3 = 6; find two numbers whose product is 6 and sum is 7, which are 1 and 6. Then rewrite and factor by grouping.

当 a ≠ 1 时，可以采用十字相乘法：先将 a 与 c 相乘（2 × 3 = 6），再找两个数使乘积为 6、和为 7，即 1 和 6。然后拆分中间项并分组因式分解。

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

Always check your answer by re-expanding. If the expansion matches the original quadratic, the factorisation is correct.

完成因式分解后务必重新展开验证，若展开结果与原式一致，说明分解正确。

4. Solving Quadratic Equations by Factorisation | 因式分解法求解

The principle behind this method is simple: if the product of two factors is zero, then at least one factor must be zero. This is called the zero product property.

这种方法的核心原理很简单：如果两个因式的乘积为零，那么至少有一个因式为零，这称为“零乘积性质”。

Worked Example: Solve x² – 5x – 14 = 0. First factorise: we need two numbers with product -14 and sum -5, which are -7 and 2. Hence (x – 7)(x + 2) = 0.

例题：解方程 x² – 5x – 14 = 0。先分解因式：需要找乘积为 -14、和为 -5 的两个数，即 -7 和 2。因此 (x – 7)(x + 2) = 0。

x – 7 = 0 or x + 2 = 0, so x = 7 or x = -2

The solution set is written as x = 7 or x = -2. Both solutions can be verified by substituting back into the original equation. Note that the word “or” is essential; a quadratic equation has at most two distinct real roots.

解集写作 x = 7 或 x = -2。将两个解分别代回原方程即可验证。注意必须使用“或”字；一元二次方程至多有两个不同的实数根。

Important warning: You may only divide both sides by x if you are certain x ≠ 0; otherwise you will lose the root x = 0. For example, x² = 5x cannot be solved by dividing by x (which gives x = 5) alone, because x = 0 is also a solution.

重要警示：只有在确定 x ≠ 0 时才可以将方程两边同时除以 x，否则会丢失 x = 0 这个根。例如，x² = 5x 不能只通过除以 x 得到 x = 5，因为 x = 0 同样是方程的解。

5. Solving by Completing the Square | 配方法求解

Completing the square rewrites a quadratic in the form (x + p)² + q. For x² + bx, add and subtract (b/2)². This method works for all quadratics, including those that cannot be factorised.

配方法将二次式改写为 (x + p)² + q 的形式。对于 x² + bx，需要加减 (b/2)²。这种方法适用于所有二次方程，包括无法因式分解的情况。

Worked Example: Solve x² + 6x + 2 = 0. Take half of 6, which is 3, and square it to get 9.

例题：解方程 x² + 6x + 2 = 0。取 6 的一半得 3，平方后得到 9。

(x + 3)² – 9 + 2 = 0, so (x + 3)² = 7

x + 3 = ±√7, so x = -3 ± √7

Remember that taking the square root of both sides introduces both the positive and negative roots, hence the ± symbol. The answer should be left in surd form unless the question asks for a decimal approximation.

注意对两边同时开平方时会出现正负两个根，因此使用 ± 符号。除非题目要求保留小数，否则答案应保留根式（无理数）形式。

When the coefficient of x² is not 1, factor it out first. For example, 2x² + 8x + 3 becomes 2(x² + 4x) + 3, then complete the square inside the bracket.

当 x² 的系数不为 1 时，先提取该系数。例如，2x² + 8x + 3 化为 2(x² + 4x) + 3，再在括号内配方。

6. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation of the form ax² + bx + c = 0. You should memorise it carefully; it is not provided on the IGCSE 0580 formula sheet in all examining board variants.

求根公式适用于所有形式为 ax² + bx + c = 0 的二次方程。请务必牢记该公式，在部分 IGCSE 0580 考局的公式表中不会给出它。

x = (-b ± √(b² – 4ac)) / 2a

Worked Example: Solve 2x² + 3x – 5 = 0 using the formula. Substituting a = 2, b = 3, c = -5:

例题：用求根公式解 2x² + 3x – 5 = 0。代入 a = 2，b = 3，c = -5：

x = (-3 ± √(9 + 40)) / 4 = (-3 ± √49) / 4 = (-3 ± 7) / 4

x = 4/4 = 1, or x = -10/4 = -2.5

Always simplify your answers fully. Check whether the question asks for exact values (surd form) or values correct to a specified number of decimal places or significant figures.

必须将答案化到最简形式。同时注意题目要求的是精确值（根式形式），还是保留指定小数位数或有效数字的近似值。

7. The Discriminant | 判别式

The expression b² – 4ac inside the square root is called the discriminant, denoted by the Greek letter Δ (delta). It tells us the nature of the roots without solving the equation fully.

根号内的表达式 b² – 4ac 称为判别式，用希腊字母 Δ（德尔塔）表示。它能在不完全求解的情况下判断方程根的性质。

Discriminant	Nature of Roots	Graph Intersection with x-axis
Δ > 0	two distinct real roots	two distinct points
Δ = 0	one repeated real root	one point (touches the axis)
Δ < 0	no real roots	does not intersect the x-axis

Example: For 3x² – 2x + 4 = 0, the discriminant is Δ = (-2)² – 4 × 3 × 4 = 4 – 48 = -44. Since Δ < 0, there are no real roots and the graph lies entirely above the x-axis (because a > 0).

示例：对于方程 3x² – 2x + 4 = 0，判别式为 Δ = (-2)² – 4 × 3 × 4 = 4 – 48 = -44。因为 Δ < 0，所以方程没有实数根，且由于 a > 0，抛物线完全位于 x 轴上方。

A common exam question type is “find the value of k such that the equation has a repeated root”. Set the discriminant equal to zero and solve for k.

常见题型是“求 k 的值，使方程有重根”。此时只需令判别式等于零，然后解关于 k 的方程即可。

8. Sketching Quadratic Graphs | 绘制二次函数图像

To sketch a quadratic graph y = ax² + bx + c, you need three key features: the y-intercept (0, c), the roots (where y = 0), and the vertex (turning point). The sign of a tells you whether the vertex is a minimum (a > 0) or a maximum (a < 0).

绘制二次函数图像 y = ax² + bx + c 需要确定三个关键特征：y 轴截距 (0, c)、根（即 y = 0 的点）以及顶点（转向点）。a 的符号决定顶点是最低点（a > 0）还是最高点（a < 0）。

The x-coordinate of the vertex is given by x = -b / 2a. Substitute this value back into the equation to find the y-coordinate.

顶点的 x 坐标为 x = -b / 2a。将其代回原方程即可求得 y 坐标。

Example: Sketch y = x² – 2x – 3. The y-intercept is (0, -3). Factorising gives (x – 3)(x + 1), so the roots are x = 3 and x = -1. The vertex has x = -(-2)/(2 × 1) = 1, and y = 1 – 2 – 3 = -4, so the vertex is (1, -4).

示例：绘制 y = x² – 2x – 3 的图像。y 截距为 (0, -3)。因式分解得 (x – 3)(x + 1)，所以根为 x = 3 和 x = -1。顶点 x = -(-2)/(2 × 1) = 1，y = 1 – 2 – 3 = -4，故顶点为 (1, -4)。

When the quadratic has no real roots (Δ < 0), the graph does not cross the x-axis. In this case, find the vertex and y-intercept, and note whether the graph is entirely above or below the x-axis based on the sign of a.

当二次式没有实数根（Δ < 0）时，图像不与 x 轴相交。此时只需找到顶点和 y 截距，并根据 a 的符号判断抛物线完全位于 x 轴上方还是下方。

9. Applying Quadratics to Word Problems | 二次方程应用题

Many IGCSE questions require setting up a quadratic equation from a real-life situation. A typical example involves area, revenue, or projectile motion. Always define your variable clearly at the start.

许多 IGCSE 考题需要从实际情境中建立二次方程，常见于面积、利润和抛体运动问题。解题时首先明确设未知数。

Worked Example: A rectangle has length (x + 5) cm and width (x – 2) cm. Its area is 40 cm². Find the value of x.

例题：一个长方形的长为 (x + 5) cm，宽为 (x – 2) cm，面积为 40 cm²。求 x 的值。

(x + 5)(x – 2) = 40 ⇒ x² + 3x – 10 = 40 ⇒ x² + 3x – 50 = 0

By the quadratic formula, x = (-3 ± √(9 + 200)) / 2 = (-3 ± √209) / 2. The positive solution is approximately 5.73, and the negative solution is rejected because a length cannot be negative.

利用求根公式，x = (-3 ± √(9 + 200)) / 2 = (-3 ± √209) / 2。正数解约为 5.73，负数解因为长度不能为负而被舍去。

When solving word problems, always check that your answers make sense in context. Reject any negative solutions for lengths, times, or numbers of items unless the question explicitly allows negative values.

在解应用题时，务必检验答案是否符合实际意义。对于长度、时间或物品数量，除非题目明确允许，否则应舍去负数解。

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

Below are the most frequently tested pitfalls and the strategies that top-scoring students use to avoid them.

以下是最常见的失分点和高分学生常用的应对策略。

Mistake: forgetting to rearrange the equation to the form ax² + bx + c = 0 before factorising or applying the formula. Always move all terms to one side first.

错误：在使用因式分解或求根公式前，忘记将方程整理为 ax² + bx + c = 0 的形式。务必先将所有项移到等号一侧。
Mistake: losing solutions when solving x² = k by writing only x = √k. The correct solution is x = ±√k.

错误：解 x² = k 时只写 x = √k，导致漏根。正确答案应为 x = ±√k。
Mistake: incorrect signs when substituting negative values into the quadratic formula. Use a calculator step-by-step and write down every substitution.

错误：将负数代入求根公式时符号出错。建议分步计算，并写出每一步的代入过程。
Tip: check factorisations by expanding. This takes five seconds and eliminates careless errors.

技巧：通过重新展开来检验因式分解是否正确。这只需五秒钟，却能避免粗心错误。
Tip: read the question for the required form. If it says “correct to 3 significant figures”, a surd answer will not receive full marks.

技巧：仔细审题，注意题目要求的答案形式。如果要求“精确到 3 位有效数字”，保留根式将无法得满分。
Tip: for sketching questions, label all intercepts and the vertex with their exact coordinates. Unlabelled graphs typically lose method marks.

技巧：作图题必须标出所有截距和顶点的精确坐标。未标注的图形通常会被扣除步骤分。

In the exam, allocate about 3 to 5 minutes per quadratic question. If a factorisation does not come to mind within 30 seconds, switch to the quadratic formula; it always works for equations with real roots.

考试中，每道二次方程题目建议分配 3 到 5 分钟。如果 30 秒内想不出因式分解的方法，应立即改用求根公式，因为它对所有有实数根的方程都有效。

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Quadratic Equations Mastery | 一元二次方程完全攻略