Quadratic Functions & Equations: Completing the Square, the Discriminant and Graphs | 二次函数与方程：配方法、判别式与图象

📚 Quadratic Functions & Equations: Completing the Square, the Discriminant and Graphs | 二次函数与方程：配方法、判别式与图象

Quadratic functions and equations are one of the most frequently tested topics in IGCSE Mathematics. From solving equations by factorisation to sketching parabolas and interpreting the discriminant, this topic connects algebra, graphs and problem solving in a single strand of questions.

二次函数与方程是 IGCSE 数学中考查频率最高的内容之一。无论是因式分解解方程、配方、绘制抛物线图像，还是通过判别式判断根的性质，这个主题都将代数、图像与实际应用紧密地串联在一起。

1. Standard Form and the Shape of the Parabola | 标准形式与抛物线形状

A quadratic function is written in the standard form y = ax² + bx + c, where a, b and c are constants and a ≠ 0. The graph of a quadratic function is always a parabola — a smooth, symmetric curve.

二次函数的标准形式为 y = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。二次函数的图像始终是一条抛物线——一条平滑且对称的曲线。

The sign of a determines the orientation of the parabola: if a > 0, the parabola opens upwards and has a minimum point; if a < 0, it opens downwards and has a maximum point. The larger the absolute value of a, the narrower the parabola.

a 的正负决定了抛物线的开口方向：当 a > 0 时，抛物线开口向上，存在最小值点；当 a < 0 时，抛物线开口向下，存在最大值点。|a| 越大，抛物线的开口越窄。

a > 0: opening upwards / 开口向上
a < 0: opening downwards / 开口向下
c: the y-intercept / c 为 y 轴截距

2. Solving Quadratic Equations by Factorisation | 因式分解法解二次方程

When a quadratic equation can be written in the form (x + p)(x + q) = 0, we can solve it immediately using the zero product property: if the product of two factors is zero, then at least one of the factors must be zero.

当二次方程可以写成 (x + p)(x + q) = 0 的形式时，我们可以直接利用零积性质求解：如果两个因式的乘积为零，那么至少有一个因式必须为零。

For example, solve x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5 — these are 2 and 3. Hence (x + 2)(x + 3) = 0, so x = −2 or x = −3.

例如，解方程 x² + 5x + 6 = 0。我们需要找到两个数，它们的乘积为 6、和为 5——这两个数是 2 和 3。因此 (x + 2)(x + 3) = 0，所以 x = −2 或 x = −3。

x² + 5x + 6 = (x + 2)(x + 3) = 0 → x = −2 or x = −3

3. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x + h)² + k. This is particularly useful for finding the vertex of a parabola and solving equations that do not factorise easily.

配方法将一个二次表达式改写为 a(x + h)² + k 的形式。这种方法特别适用于求抛物线的顶点，也适用于解那些不易因式分解的方程。

Take x² + 6x + 5. Half of 6 is 3, and 3² = 9, so we write x² + 6x + 9 − 9 + 5 = (x + 3)² − 4. The minimum value of this expression is −4, reached when x = −3.

以 x² + 6x + 5 为例。6 的一半是 3，而 3² = 9，因此我们写成 x² + 6x + 9 − 9 + 5 = (x + 3)² − 4。这个表达式的最小值为 −4，在 x = −3 时取得。

x² + 6x + 5 = (x + 3)² − 4

When the coefficient of x² is not 1, first factor out a, then complete the square inside the bracket. For example, 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5.

当 x² 的系数不为 1 时，先提取公因数 a，再在括号内配方。例如，2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5。

4. The Quadratic Formula | 求根公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula. This method works for all quadratic equations, including those that cannot be factorised.

对于任意二次方程 ax² + bx + c = 0，其解都可以由求根公式给出。这种方法适用于所有二次方程，包括那些无法因式分解的情况。

x = (−b ± √(b² − 4ac)) / 2a

For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substituting into the formula gives x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4. This simplifies to x ≈ 2.58 or x ≈ −0.58, correct to two decimal places.

例如，解方程 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入求根公式得 x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4。化简后 x ≈ 2.58 或 x ≈ −0.58，精确到两位小数。

5. The Discriminant and the Nature of Roots | 判别式与根的性质

The expression b² − 4ac is called the discriminant, often denoted by Δ. Its value determines how many real roots a quadratic equation has, without actually solving the equation.

表达式 b² − 4ac 称为判别式，通常用 Δ 表示。它的值决定了二次方程有多少个实数根，而无需真正解方程。

Value of Δ 判别式的值	Nature of Roots 根的性质	Graph Interpretation 图像含义
Δ > 0	Two distinct real roots 两个不相等的实数根	Parabola crosses the x-axis twice 抛物线与 x 轴有两个交点
Δ = 0	One repeated real root 一个相等的实数根（重根）	Parabola touches the x-axis once 抛物线与 x 轴相切
Δ < 0	No real roots 无实数根	Parabola does not intersect the x-axis 抛物线与 x 轴无交点

If the question asks for “equal roots”, this is a direct signal that Δ = 0. This type of question often involves finding an unknown constant k, and is very common in IGCSE papers.

如果题目中出现“相等的根”（equal roots），这就是一个明确信号，说明 Δ = 0。这类题目通常要求求未知常数 k 的值，在 IGCSE 考卷中非常常见。

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

Every parabola has a vertical axis of symmetry that passes through its vertex. For y = ax² + bx + c, the x-coordinate of the vertex is given by x = −b/(2a).

每条抛物线都有一条经过其顶点的竖直对称轴。对于 y = ax² + bx + c，顶点的 x 坐标为 x = −b/(2a)。

x_vertex = −b / (2a)

Once the x-coordinate is found, substitute it back into the original equation to get the y-coordinate of the vertex. Alternatively, when the function is written in completed square form y = a(x + h)² + k, the vertex is simply (−h, k).

求得 x 坐标后，将其代回原方程即可得到顶点的 y 坐标。另一种方法是，当函数写成配方法形式 y = a(x + h)² + k 时，顶点坐标直接就是 (−h, k)。

For example, the vertex of y = (x − 3)² + 2 is (3, 2), and the axis of symmetry is the line x = 3.

例如，y = (x − 3)² + 2 的顶点为 (3, 2)，对称轴为直线 x = 3。

7. Sketching the Graph of a Quadratic | 绘制二次函数图像

To sketch a quadratic graph accurately, you need to find three types of key points: the y-intercept, the x-intercepts (if they exist), and the vertex. The y-intercept is simply the value of c in y = ax² + bx + c.

为了准确地绘制二次函数图像，你需要找到三类关键点：y 轴截距、x 轴截距（如果存在）以及顶点。y 轴截距就是 y = ax² + bx + c 中 c 的值。

The x-intercepts are found by setting y = 0 and solving the resulting quadratic equation. If the discriminant is negative, the parabola does not cross the x-axis, and you only need the vertex and y-intercept for a rough sketch.

x 轴截距通过令 y = 0 并解对应的二次方程来求得。如果判别式为负，抛物线不经过 x 轴，此时只需顶点和 y 轴截距即可画出大致图像。

Step 1: find the y-intercept (0, c) / 第一步：求 y 轴截距 (0, c)
Step 2: solve ax² + bx + c = 0 for x-intercepts / 第二步：解方程求 x 轴截距
Step 3: find the vertex using x = −b/(2a) / 第三步：用 x = −b/(2a) 求顶点
Step 4: confirm the direction from the sign of a / 第四步：根据 a 的符号确定开口方向

8. Transformations of Quadratic Graphs | 二次函数图像的变换

IGCSE questions often combine quadratics with graph transformations. The completed square form y = (x + h)² + k makes transformations very easy to identify.

IGCSE 题目经常将二次函数与图像变换结合起来。配方法形式 y = (x + h)² + k 使变换规律非常容易识别。

The graph of y = x² is translated horizontally by −h and vertically by k. For example, y = (x − 2)² + 3 is the graph of y = x² shifted 2 units to the right and 3 units upwards.

y = x² 的图像沿水平方向平移 −h，沿竖直方向平移 k。例如，y = (x − 2)² + 3 就是 y = x² 的图像向右平移 2 个单位、再向上平移 3 个单位。

y = (x − h)² + k: shift right by h and up by k / 向右 h、向上 k
y = (x + h)² − k: shift left by h and down by k / 向左 h、向下 k
y = −f(x): reflection in the x-axis / 关于 x 轴对称

9. Applications in Problem Solving | 实际应用问题

Quadratic equations appear in many real-world contexts, including area problems, projectile motion, and optimisation. A common IGCSE style question involves finding the dimensions of a rectangle given its area.

二次方程出现在许多实际场景中，包括面积问题、抛体运动和最值问题。IGCSE 中常见的一类题目是：给定面积，求矩形的尺寸。

For example, a rectangle has length (x + 4) cm and width (x − 1) cm, and its area is 36 cm². We set up the equation (x + 4)(x − 1) = 36. Expanding gives x² + 3x − 40 = 0, which factorises as (x + 8)(x − 5) = 0. Since x must be positive, x = 5.

例如，一个矩形的长为 (x + 4) cm，宽为 (x − 1) cm，面积为 36 cm²。我们建立方程 (x + 4)(x − 1) = 36。展开得 x² + 3x − 40 = 0，因式分解为 (x + 8)(x − 5) = 0。由于 x 必须为正，所以 x = 5。

(x + 4)(x − 1) = 36 → x² + 3x − 40 = 0 → x = 5

Always check whether both roots make sense in the original problem. Negative lengths or times must be rejected, even if they satisfy the equation algebraically.

务必检查两个根在原始问题中是否有意义。即使负根在代数上满足方程，也必须舍去，因为长度或时间不能为负数。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students lose marks on quadratic questions for avoidable reasons. The most common mistake is forgetting to rearrange the equation into the standard form ax² + bx + c = 0 before applying the quadratic formula.

许多学生在二次函数题目中因可避免的失误而丢分。最常见的错误是忘记先将方程化为标准形式 ax² + bx + c = 0，就直接使用求根公式。

Always rearrange before solving / 求解前一定要先整理方程
Check the sign of a before sketching / 画图前先确认 a 的正负号
When using the formula, carefully substitute negative values of b / 使用求根公式时，小心代入 b 的负值
Read whether the question asks for exact values or decimals / 注意题目要求精确值还是小数
For word problems, reject impossible negative answers / 应用题中舍去不可能的负根

In the exam, show every step of your working. Even if your final answer is slightly wrong, a correct factorisation or correct substitution into the formula can still earn you method marks.

考试中请展示每一步计算过程。即使最终答案略有偏差，只要因式分解或代入求根公式的步骤正确，仍然可以获得步骤分。

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Quadratic Functions & Equations: Completing the Square, the Discriminant and Graphs | 二次函数与方程：配方法、判别式与图象