Mastering Quadratic Functions & Equations | 二次函数与方程精讲

📚 Mastering Quadratic Functions & Equations | 二次函数与方程精讲

The quadratic function and equation form one of the most important cornerstones of IGCSE Mathematics. From factorisation and the quadratic formula to completing the square and interpreting graphs, this topic appears in almost every exam paper—often worth 15–20% of the total marks. In this revision guide, we will break it down step by step, ensuring you understand both the methods and the underlying logic.

二次函数与方程是 IGCSE 数学中最核心的基石之一。无论是因式分解、求根公式、配方法，还是函数图像的解读，这一知识点几乎出现在每份试卷中，通常占总分的 15%–20%。在本复习指南中，我们将一步步拆解这些内容，帮助你既掌握方法，也理解背后的逻辑。

1. Standard Form of a Quadratic | 二次函数的标准形式

A quadratic expression is any expression of the form ax² + bx + c, where a, b and c are constants and a ≠ 0. If a = 0, the expression becomes linear. The term ax² is called the quadratic term, bx the linear term, and c the constant term.

二次表达式是指形如 ax² + bx + c 的代数式，其中 a、b、c 为常数，且 a ≠ 0。若 a = 0，表达式就退化为一次式。ax² 称为二次项，bx 称为一次项，c 称为常数项。

When we set a quadratic expression equal to zero, we obtain a quadratic equation: ax² + bx + c = 0. The values of x that satisfy this equation are called roots or solutions.

当我们将二次表达式等于零时，就得到二次方程：ax² + bx + c = 0。满足该方程的 x 值称为方程的根或解。

ax² + bx + c = 0 , a ≠ 0

2. Solving by Factorisation | 因式分解法求解

Factorisation is the fastest method when the quadratic has simple integer roots. The principle is to rewrite ax² + bx + c as a product of two linear factors, then use the zero-product property: if p × q = 0, then p = 0 or q = 0.

当二次方程具有简单的整数根时，因式分解是最快捷的方法。其原理是将 ax² + bx + c 改写为两个一次因式的乘积，然后利用零积性质：若 p × q = 0，则 p = 0 或 q = 0。

For a monic quadratic (where a = 1), we look for two numbers that multiply to give c and add to give b. For example, x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0, giving x = -2 or x = -3.

对于首项系数为 1 的二次式（a = 1），我们需要找到两个数，它们相乘等于 c，相加等于 b。例如，x² + 5x + 6 = 0 可化为 (x + 2)(x + 3) = 0，从而得到 x = -2 或 x = -3。

For a non-monic quadratic (a ≠ 1), we use the method of factorising by grouping, often called the “ac method”. Multiply a by c, find two numbers that multiply to ac and add to b, then split the middle term and factor by grouping.

对于首项系数不为 1 的二次式（a ≠ 1），我们使用分组分解法，通常称为”ac 方法”。先将 a 乘以 c，找到两个数使它们相乘等于 ac、相加等于 b，然后拆分中间项并分组因式分解。

Equation 方程	Factors 因式	Roots 根
x² + 7x + 10 = 0	(x + 2)(x + 5)	x = -2, x = -5
x² – 6x + 8 = 0	(x – 2)(x – 4)	x = 2, x = 4
2x² + 5x + 3 = 0	(2x + 3)(x + 1)	x = -1.5, x = -1

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

Completing the square rewrites a quadratic in the form a(x + p)² + q. This form is extremely useful because it directly reveals the vertex of the parabola and allows us to solve equations that do not factorise neatly.

配方法是将二次式改写为 a(x + p)² + q 的形式。这种形式极为有用，因为它直接揭示了抛物线的顶点坐标，并且允许我们求解无法简洁分解的方程。

For a monic quadratic x² + bx + c, take half of b, square it, and add and subtract that value. For example, x² + 6x + 2 = 0 becomes (x + 3)² – 9 + 2 = 0, which simplifies to (x + 3)² – 7 = 0.

对于首项系数为 1 的二次式 x² + bx + c，取 b 的一半，求其平方，然后加一个并减一个该值。例如，x² + 6x + 2 = 0 可化为 (x + 3)² – 9 + 2 = 0，即 (x + 3)² – 7 = 0。

Then solve directly: (x + 3)² = 7, so x + 3 = ±√7, hence x = -3 ± √7. This method always works, and it is also the foundation for deriving the quadratic formula.

然后直接求解：(x + 3)² = 7，因此 x + 3 = ±√7，故 x = -3 ± √7。这种方法始终有效，同时也是推导求根公式的基础。

When the coefficient a is not 1, factor it out from the first two terms first. For example, 2x² + 8x + 5 = 0 becomes 2(x² + 4x) + 5 = 2[(x + 2)² – 4] + 5 = 2(x + 2)² – 3.

当系数 a 不为 1 时，先从第一项和第二项中提取 a。例如，2x² + 8x + 5 = 0 变为 2(x² + 4x) + 5 = 2[(x + 2)² – 4] + 5 = 2(x + 2)² – 3。

4. The Quadratic Formula | 求根公式

The quadratic formula is the most universal method for solving any quadratic equation. It is derived by completing the square on the general form ax² + bx + c = 0, and it always gives the exact solutions.

求根公式是解任何二次方程的最通用方法。它通过对一般形式 ax² + bx + c = 0 配方推导而来，并且始终给出精确解。

x = (-b ± √(b² – 4ac)) / 2a

To use the formula, identify the values of a, b and c from your equation, substitute them into the formula, and simplify carefully. Pay close attention to negative signs—one of the most common mistakes is mishandling a negative b or c.

使用公式时，先确定方程中 a、b、c 的值，代入公式并仔细化简。要特别注意负号——最常见的错误之一是错误处理负的 b 或 c。

For example, in the equation 2x² – 4x – 6 = 0, we have a = 2, b = -4, c = -6. Substituting gives x = (4 ± √(16 + 48)) / 4 = (4 ± 8) / 4, so x = 3 or x = -1.

例如，在方程 2x² – 4x – 6 = 0 中，a = 2，b = -4，c = -6。代入得 x = (4 ± √(16 + 48)) / 4 = (4 ± 8) / 4，因此 x = 3 或 x = -1。

5. The Discriminant | 判别式

The expression b² – 4ac inside the quadratic formula is called the discriminant. It tells us the nature of the roots without actually solving the equation.

求根公式中的表达式 b² – 4ac 称为判别式。它无需实际求解方程就能告诉我们根的性质。

If b² – 4ac > 0, the equation has two distinct real roots. If b² – 4ac = 0, the equation has exactly one repeated real root (also called a double root). If b² – 4ac < 0, the equation has no real roots—the solutions are complex numbers, which are not covered at IGCSE level.

若 b² – 4ac > 0，方程有两个不相等的实数根。若 b² – 4ac = 0，方程恰有一个重复的实数根（也称为二重根）。若 b² – 4ac < 0，方程没有实数根——解为复数，这在 IGCSE 阶段不涉及。

Discriminant 判别式	Nature of Roots 根的性质	Graph 图像
b² – 4ac > 0	Two distinct real roots 两个不同实数根	Crosses x-axis twice 与 x 轴有两个交点
b² – 4ac = 0	One repeated root 一个二重根	Touches x-axis once 与 x 轴相切
b² – 4ac < 0	No real roots 无实数根	Does not meet x-axis 不与 x 轴相交

6. Graphs of Quadratic Functions | 二次函数的图像

The graph of a quadratic function is a smooth curve called a parabola. If a > 0, the parabola opens upwards (a “smile” shape). If a < 0, it opens downwards (a "frown" shape). The larger the magnitude of a, the narrower the parabola.

二次函数的图像是一条平滑的曲线，称为抛物线。若 a > 0，抛物线开口向上（呈”微笑”状）；若 a < 0，抛物线开口向下（呈"沮丧"状）。a 的绝对值越大，抛物线越窄。

There are three key features to identify on a quadratic graph: the roots (where the curve crosses the x-axis), the y-intercept (where the curve crosses the y-axis), and the vertex (the maximum or minimum point).

二次函数图像有三个关键特征：根（曲线与 x 轴的交点）、y 截距（曲线与 y 轴的交点）以及顶点（最大值或最小值点）。

The y-intercept is always simply the constant term c. The vertex can be found by completing the square: if y = a(x + p)² + q, then the vertex is at (-p, q). Alternatively, the x-coordinate of the vertex is -b/(2a).

y 截距始终就是常数项 c。顶点可以通过配方法求得：若 y = a(x + p)² + q，则顶点坐标为 (-p, q)。或者，顶点的 x 坐标为 -b/(2a)。

7. Sketching Quadratics | 二次函数图像的画法

Sketching a quadratic requires finding two or three key points. The standard approach is: first find the y-intercept by setting x = 0; then find the roots by setting y = 0; finally find the vertex using completing the square or the formula x = -b/(2a).

画二次函数草图需要找两三个关键点。标准步骤是：先令 x = 0 求 y 截距；再令 y = 0 求根；最后通过配方或公式 x = -b/(2a) 求顶点。

Once these points are plotted, draw a smooth symmetric curve through them. Remember that the parabola has an axis of symmetry passing vertically through the vertex, so the curve on one side of the vertex is a mirror image of the other side.

标出这些点之后，绘制一条平滑且对称的曲线穿过它们。请记住，抛物线有一条经过顶点的竖直对称轴，因此顶点一侧的曲线是另一侧的镜像。

If the equation has no real roots, the curve does not cross the x-axis. In this case, plot the y-intercept, the vertex, and one additional point on each side of the axis of symmetry to get an accurate sketch.

若方程没有实数根，曲线不与 x 轴相交。在这种情况下，画出 y 截距、顶点以及对称轴两侧各一个额外点，以获得准确的草图。

8. Solving Quadratic Inequalities | 二次不等式求解

A quadratic inequality looks like ax² + bx + c > 0 or ax² + bx + c < 0. The solution set is best understood graphically: we ask where the parabola lies above or below the x-axis.

二次不等式形如 ax² + bx + c > 0 或 ax² + bx + c < 0。其解集最好通过图像来理解：我们关注抛物线在 x 轴上方或下方的区间。

First solve the corresponding equation to find the roots. Then sketch a quick graph to see which intervals satisfy the inequality. For an upwards-opening parabola with roots r₁ and r₂:

先解对应方程求出根。然后快速画图判断哪些区间满足不等式。对于开口向上且有根 r₁ 和 r₂ 的抛物线：

If ax² + bx + c > 0, the solution is x < r₁ or x > r₂ 若 ax² + bx + c > 0，解为 x < r₁ 或 x > r₂
If ax² + bx + c < 0, the solution is r₁ < x < r₂ 若 ax² + bx + c < 0，解为 r₁ < x < r₂

Be careful with strict inequalities (>) versus non-strict ones (≥ or ≤). Strict inequalities use open intervals, while non-strict ones include the roots themselves.

注意严格不等式（>）与非严格不等式（≥ 或 ≤）的区别。严格不等式使用开区间，而非严格不等式则包含根本身。

9. Applications: Word Problems | 应用：文字题

Quadratic equations are widely used to model real-world situations. Common IGCSE problems involve area, projectile motion, number relationships, and profit maximisation. The key is to translate the words into a quadratic equation first.

二次方程广泛用于建模现实情境。IGCSE 常见的应用题涉及面积、抛体运动、数之间的关系以及利润最大化。关键是将文字首先转化为二次方程。

For example, consider the problem: “A rectangular garden has a length that is 3 metres longer than its width. Its area is 28 square metres. Find its width.” If the width is x, then the length is x + 3, so x(x + 3) = 28, which expands to x² + 3x – 28 = 0.

例如：”一个矩形花园的长比宽多 3 米，面积为 28 平方米，求宽。” 设宽为 x，则长为 x + 3，因此 x(x + 3) = 28，展开得 x² + 3x – 28 = 0。

Factorising gives (x + 7)(x – 4) = 0, so x = -7 or x = 4. Since width cannot be negative, the width is 4 metres. Always check whether your solutions make sense in the context of the problem.

因式分解得 (x + 7)(x – 4) = 0，故 x = -7 或 x = 4。由于宽度不能为负，因此宽为 4 米。务必检查解是否在问题情境中合理。

10. The Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, there are elegant relationships that allow us to write new equations or check our answers:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，存在优美的关系式，可以帮助我们构造新方程或检验答案：

Sum of roots: α + β = -b/a

Product of roots: αβ = c/a

These relationships are derived directly from the factorised form. If you know the sum S and the product P of the roots, you can construct the equation x² – Sx + P = 0 (for a = 1).

这些关系直接从因式分解形式推导而来。如果你知道根的和 S 与积 P，就可以构造方程 x² – Sx + P = 0（当 a = 1 时）。

For example, if a quadratic has roots 2 and -5, then S = -3 and P = -10, giving the equation x² + 3x – 10 = 0. This is a powerful tool for solving certain problems quickly.

例如，若某二次方程的根为 2 和 -5，则 S = -3，P = -10，得到方程 x² + 3x – 10 = 0。这是快速解决某些问题的有力工具。

11. Common Mistakes to Avoid | 常见错误避坑指南

Many students lose marks on quadratic questions due to a few recurring mistakes. Being aware of these pitfalls is the first step to avoiding them.

许多学生在二次函数相关的题目上因几个反复出现的错误而失分。意识到这些陷阱是避免它们的第一步。

Forgetting to set the equation to zero before factorising — always rearrange to ax² + bx + c = 0 first. 因式分解前忘记将方程化为零——务必先把等式整理为 ax² + bx + c = 0。
Misapplying the quadratic formula when b is negative — double-check your substitutions. 当 b 为负数时错误使用求根公式——请反复检查代入过程。
Dropping the ± symbol when taking square roots — always include both positive and negative solutions. 开平方时漏掉 ± 符号——始终同时保留正负两个解。
Confusing the x-coordinate of the vertex with the axis of symmetry — they are actually the same line. 混淆顶点的 x 坐标与对称轴——事实上它们是同一条线。
Discarding a negative solution without justification in word problems — only discard it if the context demands a positive value. 在文字题中无故舍弃负解——只有当情境要求正值时才可舍弃。

12. Exam Strategy & Practice Tips | 考试策略与练习建议

In the IGCSE exam, quadratic questions often appear in both Paper 2 (short questions) and Paper 4 (long questions). In short-answer questions, factorisation is usually fastest. In extended, multi-part questions, you may need to interpret graphs, complete the square, and use the discriminant in sequence.

在 IGCSE 考试中，二次函数相关题目通常同时出现在 Paper 2（简答题）和 Paper 4（长答题）中。在简答题中，因式分解通常最快。在扩展的多步骤题目中，你可能需要依次解读图像、配方和使用判别式。

Here is a recommended checking procedure: after solving a quadratic equation, substitute your answers back into the original equation to verify them. This takes only a few seconds and can catch arithmetic errors before you move on.

这里有一个推荐的检验流程：解完二次方程后，将答案代回原方程进行验证。这只需几秒种，可以在继续做题之前发现运算错误。

Aim to master all three solving methods—factorisation, completing the square, and the quadratic formula. Some questions explicitly require a specific method, and fluency in all three gives you flexibility in choosing the most efficient path.

目标是掌握三种求解方法——因式分解、配方法和求根公式。有些题目会明确要求特定方法，而熟练三种方法能让你灵活选择最有效的路径。

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Mastering Quadratic Functions & Equations | 二次函数与方程精讲