IGCSE Maths Revision: Quadratic Equations & Graphs | IGCSE数学复习：二次方程与图像

📚 IGCSE Maths Revision: Quadratic Equations & Graphs | IGCSE数学复习：二次方程与图像

Quadratic equations and their graphs form one of the most important topics in IGCSE Mathematics. From solving equations to sketching parabolas, mastering this area unlocks success in algebra, coordinate geometry, and even calculus at higher levels. This revision guide covers everything you need to know, with clear explanations, worked examples, and exam-style tips.

二次方程及其图像是IGCSE数学中最重要的话题之一。从解方程到绘制抛物线，掌握这一领域将为代数、坐标几何乃至更高级别的微积分学习铺平道路。本复习指南涵盖你所需的一切：清晰的概念讲解、典型例题和考试技巧。

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

A quadratic equation is any equation that can be written in the general form ax² + bx + c = 0, where a, b, and c are constants, and a cannot be zero. The coefficient a is called the leading coefficient; it determines the direction and steepness of the parabola. The terms bx and c are often called the linear term and constant term respectively.

二次方程是指可以写成 ax² + bx + c = 0 这种一般形式的方程，其中a、b、c为常数，且a不能等于零。系数a称为首项系数，它决定抛物线的开口方向和陡峭程度。bx项和c项分别称为一次项和常数项。

For example, 2x² − 3x + 1 = 0 is quadratic, while x³ + 2x = 0 is not. Notice that x² − 5x = 0 is quadratic even though c = 0; and 3x² − 9 = 0 is quadratic even though b = 0. The only strict requirement is that the x² term must exist.

例如，2x² − 3x + 1 = 0 是二次方程，而 x³ + 2x = 0 不是。注意 x² − 5x = 0 尽管c = 0，它仍是二次方程；3x² − 9 = 0 尽管b = 0，也仍是二次方程。唯一严格的要求是必须存在x²项。

ax² + bx + c = 0, where a ≠ 0

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic has simple integer roots. The idea is to rewrite ax² + bx + c as a product of two brackets, then use the zero-product property: if AB = 0, then A = 0 or B = 0. For equations where a = 1, look for two numbers that multiply to give c and add to give b.

因式分解通常是在二次方程有简单整数根时最快的方法。其核心是将 ax² + bx + c 改写成两个括号的乘积，然后运用零积性质：若 AB = 0，则 A = 0 或 B = 0。当 a = 1 时，找两个数，它们相乘等于c，相加等于b。

Example: Solve x² − 5x + 6 = 0. We need two numbers that multiply to 6 and add to −5. These are −2 and −3. Thus (x − 2)(x − 3) = 0, giving x = 2 or x = 3.

例：解方程 x² − 5x + 6 = 0。我们需要两个数，相乘等于6，相加等于−5。这两个数是−2和−3。于是 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。

When a ≠ 1, you can use the “pair” method. Multiply a and c, then find two numbers that multiply to ac and add to b. Split the middle term, then factor by grouping.

当 a ≠ 1 时，可以使用“拆分”法。先计算 a 与 c 的乘积 ac，再找两个数相乘等于 ac 且相加等于 b。然后拆分中间项，再分组因式分解。

Example: Solve 2x² + 7x + 3 = 0. Here ac = 6, and two numbers multiplying to 6 and adding to 7 are 1 and 6. Rewrite as 2x² + x + 6x + 3 = 0. Group: x(2x + 1) + 3(2x + 1) = 0. Factor: (2x + 1)(x + 3) = 0. Hence x = −½ or x = −3.

例：解方程 2x² + 7x + 3 = 0。这里 ac = 6，两个数相乘等于6且相加等于7的数是1和6。改写为 2x² + x + 6x + 3 = 0。分组：x(2x + 1) + 3(2x + 1) = 0。提取公因式：(2x + 1)(x + 3) = 0。因此 x = −½ 或 x = −3。

3. Solving by Completing the Square | 配方法

Completing the square transforms x² + bx into a perfect square plus a constant. The rule is simple: take half the coefficient of x, square it, and add and subtract that value. For the general equation x² + bx + c = 0, we write (x + b/2)² − (b/2)² + c = 0.

配方法将 x² + bx 转化为完全平方加上一个常数的形式。规则很简单：取x系数的一半并平方，然后先加再减这个值。对于一般方程 x² + bx + c = 0，我们写成 (x + b/2)² − (b/2)² + c = 0。

Example: Solve x² + 6x + 2 = 0 by completing the square. Half of 6 is 3; square is 9. So x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7. Setting equal to 0: (x + 3)² = 7, so x + 3 = ±√7, giving x = −3 ± √7.

例：用配方法解 x² + 6x + 2 = 0。6的一半是3；平方为9。所以 x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。令它等于0：(x + 3)² = 7，因此 x + 3 = ±√7，得 x = −3 ± √7。

If a ≠ 1, first divide the whole equation by a. Completing the square not only solves equations but also reveals the turning point of the parabola, which we will see later.

如果 a ≠ 1，先将整个方程除以a。配方法不仅能解方程，还能揭示抛物线的顶点坐标，这一点我们将在后面看到。

4. The Quadratic Formula | 求根公式

The quadratic formula is the most universal solving tool. For ax² + bx + c = 0, the solutions are given by x = (−b ± √(b² − 4ac)) / (2a). This formula works for every quadratic equation, whether the roots are rational, irrational, or complex.

求根公式是最通用的求解工具。对于 ax² + bx + c = 0，解为 x = (−b ± √(b² − 4ac)) / (2a)。这个公式适用于所有二次方程，无论根是有理数、无理数还是复数。

x = (−b ± √(b² − 4ac)) / 2a

Example: Solve 3x² − 5x − 2 = 0 using the formula. Here a = 3, b = −5, c = −2. The discriminant b² − 4ac = 25 + 24 = 49. So x = (5 ± 7) / 6, giving x = 2 or x = −1/3.

例：用公式解 3x² − 5x − 2 = 0。这里 a = 3，b = −5，c = −2。判别式 b² − 4ac = 25 + 24 = 49。因此 x = (5 ± 7) / 6，得到 x = 2 或 x = −1/3。

Always write the formula clearly and substitute carefully. Common errors include sign mistakes with negative b, and forgetting to divide by 2a. Use brackets on a calculator to avoid arithmetic slips.

务必清晰书写公式并仔细代入。常见错误包括负b的符号错误，以及忘记除以2a。在计算器上使用括号以避免运算失误。

5. The Discriminant Δ | 判别式Δ

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

判别式是求根公式中根号内的表达式 Δ = b² − 4ac。它告诉我们二次方程有多少个实数根，而无需完整求解。

If Δ > 0: the equation has two distinct real roots. The graph cuts the x-axis at two points.
如果 Δ > 0：方程有两个不相等的实数根。图像与x轴相交于两个点。
If Δ = 0: the equation has exactly one real root (a repeated root). The graph touches the x-axis at the vertex.
如果 Δ = 0：方程只有一个实数根（重根）。图像在顶点处与x轴相切。
If Δ < 0: there are no real roots. The graph does not intersect the x-axis at all.
如果 Δ < 0：没有实数根。图像与x轴没有任何交点。

Example: For x² − 4x + 4 = 0, Δ = 16 − 16 = 0. The equation has one repeated root, x = 2. The parabola y = x² − 4x + 4 touches the x-axis at the point (2, 0).

例：对于 x² − 4x + 4 = 0，Δ = 16 − 16 = 0。方程有一个重根 x = 2。抛物线 y = x² − 4x + 4 在点 (2, 0) 处与x轴相切。

6. Graphing Quadratics: Parabolas | 二次函数图像：抛物线

The graph of y = ax² + bx + c is a smooth U-shaped curve called a parabola. If a > 0, the parabola opens upwards (like a “smile”). If a < 0, it opens downwards (like a "frown"). The larger the absolute value of a, the narrower the parabola.

y = ax² + bx + c 的图像是一条光滑的U形曲线，称为抛物线。如果 a > 0，抛物线开口向上（像“笑脸”）。如果 a < 0，抛物线开口向下（像“哭脸”）。|a| 越大，抛物线越窄。

To sketch a quadratic graph accurately, you need to find three key features: the y-intercept, the x-intercepts (roots), and the vertex (turning point). The y-intercept is simply c, the point (0, c). The x-intercepts are found by solving ax² + bx + c = 0.

要准确画出二次函数图像，你需要找出三个关键特征：y截距、x截距（根）和顶点。y截距就是 c，即点 (0, c)。x截距通过解 ax² + bx + c = 0 得到。

If the roots are real, mark them on the x-axis. If there are no real roots, the parabola does not cross the x-axis; it lies entirely above or below it, depending on the sign of a.

如果根是实数，在x轴上标出它们。如果没有实数根，抛物线不与x轴相交；它完全位于x轴上方或下方，取决于a的符号。

7. The Vertex and Axis of Symmetry | 顶点与对称轴

The vertex is the highest or lowest point of the parabola. Its x-coordinate is given by x = −b/(2a). Substituting this value back into the equation gives the y-coordinate. The vertical line x = −b/(2a) is the axis of symmetry of the parabola.

顶点是抛物线的最高点或最低点。其x坐标由 x = −b/(2a) 给出。将这个值代回原方程即可得到y坐标。竖直线 x = −b/(2a) 是抛物线的对称轴。

x_vertex = −b / 2a

Example: For y = 2x² + 8x + 5, a = 2, b = 8. The axis of symmetry is x = −8/(4) = −2. Substitute x = −2: y = 2(−2)² + 8(−2) + 5 = 8 − 16 + 5 = −3. So the vertex is (−2, −3). Since a > 0, this is a minimum point.

例：对于 y = 2x² + 8x + 5，a = 2，b = 8。对称轴为 x = −8/4 = −2。代入 x = −2：y = 2(−2)² + 8(−2) + 5 = 8 − 16 + 5 = −3。因此顶点为 (−2, −3)。因为 a > 0，这是一个最小值点。

An alternative method is to complete the square. Writing y = a(x − h)² + k reveals the vertex directly as (h, k). For example, y = (x − 3)² + 4 has vertex (3, 4).

另一种方法是配方法。将 y = a(x − h)² + k 的形式可直接看出顶点为 (h, k)。例如，y = (x − 3)² + 4 的顶点为 (3, 4)。

8. Sketching a Quadratic Graph | 绘制二次函数草图

Follow these steps to sketch y = ax² + bx + c quickly and correctly:

按照以下步骤快速、正确地绘制 y = ax² + bx + c 的草图：

Step 1: Identify the shape — upward if a > 0, downward if a < 0.
步骤1：确定开口方向 — 若 a > 0 向上，若 a < 0 向下。
Step 2: Find the y-intercept (0, c) and mark it on the y-axis.
步骤2：求y截距 (0, c) 并在y轴上标出。
Step 3: Solve ax² + bx + c = 0 to find the x-intercepts. Mark them if they exist.
步骤3：解 ax² + bx + c = 0 求x截距。如果存在，在x轴上标出。
Step 4: Find the vertex using x = −b/(2a), and mark it.
步骤4：用 x = −b/(2a) 求顶点并标出。
Step 5: Draw the smooth curve through these points, respecting the axis of symmetry.
步骤5：穿过这些点画出平滑曲线，保持关于对称轴对称。

Check your sketch by testing one extra point, such as x = 1. If the curve looks too wide or too narrow, check the value of a. A negative c shifts the curve down; a positive c shifts it up.

通过检验一个额外点（如 x = 1）来验证你的草图。如果曲线看起来过宽或过窄，请检查a的值。c为负时图像下移，c为正时图像上移。

9. Solving Quadratic Inequalities | 二次不等式

A quadratic inequality such as x² − x − 6 < 0 asks for the range of x where the parabola lies below the x-axis. First solve the corresponding equation x² − x − 6 = 0, which factorises as (x − 3)(x + 2) = 0, giving roots x = 3 and x = −2.

二次不等式（如 x² − x − 6 < 0）要求找出抛物线位于x轴下方时x的取值范围。首先解对应的方程 x² − x − 6 = 0，因式分解为 (x − 3)(x + 2) = 0，得到根 x = 3 和 x = −2。

Since the parabola opens upward, the region below the x-axis lies between the two roots. Hence the solution is −2 < x < 3. For the inequality ≥ 0, the solution would be x ≤ −2 or x ≥ 3.

由于抛物线开口向上，位于x轴下方的区域在两根之间。因此解为 −2 < x < 3。对于不等式 ≥ 0，解为 x ≤ −2 或 x ≥ 3。

x² − x − 6 < 0 ⇒ −2 < x < 3

Always sketch the graph first or at least visualise it in your mind. This prevents sign errors when writing the final interval. Remember: “between the roots” for a < 0 inequality when a > 0, and “outside the roots” for a > 0 inequality.

务必先画图或在脑中想象图像。这可以避免写出最终区间时出现符号错误。记住：当 a > 0 时，小于号取“两根之间”，大于号取“两根之外”。

10. Applications: Word Problems | 应用：文字应用题

Quadratic equations appear in many real-world contexts. Common IGCSE problems involve projectile motion, area of rectangles, and number puzzles. The key skill is translating a word problem into a quadratic equation and then solving it.

二次方程出现在许多现实情境中。IGCSE常见题型包括抛体运动、矩形面积和数字谜题。关键技能是将文字题转化为二次方程并求解。

Example: A rectangle has length (x + 4) cm and width (x − 1) cm. Its area is 42 cm². Find x. The area gives (x + 4)(x − 1) = 42, so x² + 3x − 4 = 42, hence x² + 3x − 46 = 0. Using the formula, x = (−3 ± √(9 + 184))/2 = (−3 ± √193)/2. Only the positive root ≈ 5.45 cm is valid, since a length cannot be negative.

例：一个矩形的长为 (x + 4) cm，宽为 (x − 1) cm，面积为 42 cm²。求x。面积公式给出 (x + 4)(x − 1) = 42，即 x² + 3x − 4 = 42，所以 x² + 3x − 46 = 0。用求根公式 x = (−3 ± √(9 + 184))/2 = (−3 ± √193)/2。只有正根 ≈ 5.45 cm 是合理的，因为长度不能为负。

For projectile motion, the height h(t) = −gt² + vt + h₀ is a quadratic in time t. The maximum height occurs at the vertex, and the time when the object hits the ground is found by setting h(t) = 0 and solving the quadratic.

对于抛体运动，高度 h(t) = −gt² + vt + h₀ 是时间t的二次函数。最大高度出现在顶点处，物体落地的时间通过令 h(t) = 0 解二次方程得到。

11. Exam Tips and Common Mistakes | 考试技巧与常见错误

Many students lose marks on quadratic topics due to avoidable errors. Here is a checklist of common pitfalls and how to avoid them:

许多学生在二次方程部分因可避免的错误而丢分。以下是常见陷阱清单及相应的规避方法：

Mistake \| 常见错误	Correction \| 纠正方法
Forgetting to put the equation in standard form before factorising.	Always rearrange to ax² + bx + c = 0 first.
错误地直接将非标准形式因式分解。	务必先将方程整理为 ax² + bx + c = 0 的标准形式。
Sign errors when using the quadratic formula, especially with −b.	Rewrite the formula with brackets: x = (−b ± √(b² − 4ac)) / (2a).
在使用求根公式时出现符号错误，尤其是 −b 的部分。	书写公式时使用括号：x = (−b ± √(b² − 4ac)) / (2a)。
Confusing the roots of the equation with the vertex of the parabola.	Roots are where y = 0; the vertex is the turning point x = −b/(2a).
将方程的根与抛物线的顶点混淆。	根是 y = 0 处的x值；顶点是转折点 x = −b/(2a)。

Always check your answers by substituting them back into the original equation. In an exam, if you get two answers but one does not make sense in the context, discard it — but mention why you are discarding it to show clear reasoning.

务必通过代回原方程来检验答案。在考试中，如果你得到两个答案但其中一个在语境中不合理，请舍去——同时说明舍去原因以展示清晰的推理过程。

12. Practice Questions | 练习题目

Try these questions on your own, then check your workings:

请独立完成以下练习，再对照你的解题过程：

Solve x² − 7x + 10 = 0 by factorisation.
用因式分解法解 x² − 7x + 10 = 0。
Solve 2x² + 5x − 3 = 0 using the quadratic formula.
用求根公式解 2x² + 5x − 3 = 0。
Write x² + 8x + 5 in the form (x + p)² + q.
将 x² + 8x + 5 写成 (x + p)² + q 的形式。
Find the vertex and axis of symmetry of y = x² − 6x + 11.
求 y = x² − 6x + 11 的顶点和对称轴。
Sketch the graph of y = x² − x − 2, labelling all intercepts and the vertex.
画出 y = x² − x − 2 的图像，标出所有截距和顶点。

Answers: 1) x = 2 or x = 5 2) x = ½ or x = −3 3) (x + 4)² − 11 4) Vertex (3, 2), axis x = 3 5) Intercepts (2,0), (−1,0), (0,−2), vertex (½, −2¼).

答案：1) x = 2 或 x = 5 2) x = ½ 或 x = −3 3) (x + 4)² − 11 4) 顶点 (3, 2)，对称轴 x = 3 5) 截距 (2,0)、(−1,0)、(0,−2)，顶点 (½, −2¼)。

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IGCSE Maths Revision: Quadratic Equations & Graphs | IGCSE数学复习：二次方程与图像