A-Level Mathematics: Quadratic Equations and Functions | A-Level 数学：二次方程与函数

📚 A-Level Mathematics: Quadratic Equations and Functions | A-Level 数学：二次方程与函数

Quadratic equations and functions form the heart of CIE A-Level Mathematics, appearing in pure mathematics, coordinate geometry, and calculus. Mastering this topic gives you the algebraic tools needed for higher-level problems, including series, exponentials, and integration.

二次方程与函数是 CIE A-Level 数学的核心内容，出现在纯数学、坐标几何和微积分中。掌握这一主题，能为后续处理级数、指数函数和积分等问题奠定代数基础。

1. Standard Form and Completing the Square | 标准形式与配方法

A quadratic equation is of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The corresponding quadratic function is f(x) = ax² + bx + c.

二次方程的标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。对应的二次函数为 f(x) = ax² + bx + c。

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is one of the most important algebraic techniques in CIE A-Level Mathematics because it gives the turning point immediately.

配方法将二次式改写为 a(x + p)² + q 的形式。这是 CIE A-Level 数学中最重要的代数技巧之一，因为可以直接看出图像的顶点（极值点）。

ax² + bx + c = a(x + b/(2a))² + (4ac − b²)/(4a)

For example, x² + 6x + 5 = (x + 3)² − 4. Here the turning point is (−3, −4), and the minimum value of the function is −4.

例如，x² + 6x + 5 = (x + 3)² − 4。这里顶点为 (−3, −4)，函数的最小值为 −4。

You should also use completing the square to solve equations, especially when the quadratic does not factorise with simple integers:

你还需要用配方法来解方程，尤其是当二次式不能用简单整数因式分解时：

x² + 4x − 1 = 0 → (x + 2)² − 5 = 0 → x = −2 ± √5

The technique also helps you sketch graphs, find inverse functions, and evaluate integrals involving quadratics.

配方法还能帮助你画函数图像、求反函数，以及计算涉及二次式的积分。

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

Factorisation relies on the null factor law: if the product of two expressions is zero, then at least one of the expressions must be zero.

因式分解法的依据是“零因子律”：如果两个表达式的乘积为零，则至少其中一个表达式必须为零。

To solve x² − 7x + 12 = 0, first factorise:

解 x² − 7x + 12 = 0，首先因式分解：

(x − 3)(x − 4) = 0

Therefore x − 3 = 0 or x − 4 = 0, so x = 3 or x = 4.

因此 x − 3 = 0 或 x − 4 = 0，所以 x = 3 或 x = 4。

Common factorisation patterns include:

常见的因式分解类型包括：

Common factor: x² − 5x = x(x − 5)
公因式：x² − 5x = x(x − 5)
Difference of squares: x² − 9 = (x − 3)(x + 3)
平方差：x² − 9 = (x − 3)(x + 3)
Trinomial: 2x² + 5x − 3 = (2x − 1)(x + 3)
三项式：2x² + 5x − 3 = (2x − 1)(x + 3)

If a quadratic contains a common factor first, always take it out before trying other factorisation methods.

如果二次式含有公因式，一定要先提出来，再尝试其他因式分解方法。

3. The Quadratic Formula and the Discriminant | 求根公式与判别式

The quadratic formula solves any quadratic equation ax² + bx + c = 0, even when factorisation is difficult or impossible.

求根公式可以解任意二次方程 ax² + bx + c = 0，即使因式分解困难甚至无法分解时也适用。

x = (−b ± √(b² − 4ac)) / (2a)

The expression b² − 4ac is called the discriminant, usually denoted by Δ (the Greek letter Delta).

表达式 b² − 4ac 称为判别式，通常用希腊字母 Δ 表示。

The discriminant is not just part of a formula; it tells you the number and type of roots before you solve the equation.

判别式不仅是求根公式的一部分；它还能让你在解方程之前判断根的个数和类型。

Δ = b² − 4ac

For example, solve 2x² − 4x + 1 = 0 using the formula:

例如，用求根公式解 2x² − 4x + 1 = 0：

x = (4 ± √(16 − 8)) / 4 = (4 ± √8) / 4 = 1 ± √2 / 2

You should always check whether the discriminant is a perfect square because if it is, the original quadratic might factorise neatly.

你应经常检查判别式是否为完全平方数；如果是，原二次式可能可以整齐地因式分解。

4. The Nature of Roots | 根的性质

For a quadratic equation ax² + bx + c = 0, the value of the discriminant determines the nature of the roots.

对于二次方程 ax² + bx + c = 0，判别式的值决定了根的性质。

Discriminant	Nature of Roots	Graph Intersections with x-axis
Δ > 0	Two real and distinct roots	Two distinct x-intercepts
Δ = 0	Two equal real roots / one repeated root	One x-intercept (touching point)
Δ < 0	No real roots	Graph does not cross the x-axis

In Chinese:

用中文表示：

判别式	根的性质	图像与 x 轴交点
Δ > 0	两个不相等的实数根	两个不同的 x 轴交点
Δ = 0	两个相等实数根 / 一个重根	一个 x 轴交点（相切）
Δ < 0	没有实数根	图像不与 x 轴相交

If the coefficients are rational and Δ is a perfect square, the roots are rational and the quadratic can be factorised using integers.

如果系数为有理数且 Δ 是完全平方数，则根是有理数，该二次式可以用整数因式分解。

5. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the following relations hold:

对于二次方程 ax² + bx + c = 0，设根为 α 和 β，则有：

α + β = −b/a and αβ = c/a

These relations let you construct a quadratic equation when the roots are known. If the roots are α and β, then the equation is:

这些关系可以帮助你在已知根时构造二次方程。若根为 α 和 β，则方程为：

x² − (α + β)x + αβ = 0

For example, a quadratic whose roots are 2 and −5 has sum −3 and product −10, so it is x² + 3x − 10 = 0.

例如，根为 2 和 −5 的二次方程中，根的和为 −3，积为 −10，所以方程为 x² + 3x − 10 = 0。

In CIE exam questions, you may be asked to find expressions such as α² + β² or 1/α + 1/β. Always rewrite these using α + β and αβ:

在 CIE 考试中，你可能会被要求计算 α² + β² 或 1/α + 1/β 等表达式的值。务必把它们改写成 α + β 与 αβ 的形式：

α² + β² = (α + β)² − 2αβ and 1/α + 1/β = (α + β)/(αβ)

6. Quadratic Graphs and Their Properties | 二次函数图像与性质

The graph of a quadratic function is a parabola. Its shape is controlled by the coefficient a.

二次函数的图像是抛物线，其开口方向由系数 a 控制。

If a > 0, the parabola opens upwards and the vertex is a minimum point.
如果 a > 0，抛物线开口向上，顶点为最小值点。
If a < 0, the parabola opens downwards and the vertex is a maximum point.
如果 a < 0，抛物线开口向下，顶点为最大值点。

To sketch a quadratic graph, you need three key features:

要画出二次函数图像的草图，你需要三个关键特征：

the vertex, found by completing the square;
顶点，通过配方法求得；
the y-intercept, which is c in y = ax² + bx + c;
y 轴截距，即 y = ax² + bx + c 中的 c；
the x-intercepts, found by solving ax² + bx + c = 0.
x 轴截距，通过解 ax² + bx + c = 0 求得。

The axis of symmetry is the vertical line through the vertex:

对称轴是经过顶点的竖直直线：

x = −b/(2a)

If there are no real roots, the curve never touches the x-axis; it lies entirely above the x-axis when a > 0 and Δ < 0.

如果没有实数根，曲线不会接触 x 轴；当 a > 0 且 Δ < 0 时，整条曲线都在 x 轴上方。

7. Maximum and Minimum Values | 最大值与最小值

Completing the square is the cleanest way to find the maximum or minimum value of a quadratic function.

配方法是求二次函数最大值或最小值最简洁的方法。

f(x) = a(x − h)² + k

The vertex is (h, k). If a > 0, k is the minimum value of f. If a < 0, k is the maximum value of f.

顶点为 (h, k)。若 a > 0，则 k 是 f 的最小值；若 a < 0，则 k 是 f 的最大值。

Example: f(x) = −x² + 4x + 1. Complete the square:

例：f(x) = −x² + 4x + 1。配方法如下：

f(x) = −(x² − 4x) + 1 = −(x − 2)² + 5

Therefore the maximum value is 5, which occurs at x = 2. The range of f is y ≤ 5.

因此最大值为 5，在 x = 2 处取得。函数 f 的值域为 y ≤ 5。

This idea is often combined with domain restrictions. For example, on 0 ≤ x ≤ 4, you must check the endpoint values as well as the vertex.

这一思想常与定义域限制结合。例如在 0 ≤ x ≤ 4 上求最值时，除了顶点之外，还必须检查端点处函数值。

8. Solving Quadratic Inequalities | 二次不等式

To solve a quadratic inequality, first rearrange it so that one side is zero, then factorise or use the quadratic formula.

解二次不等式时，先把一端化为零，然后因式分解或使用求根公式。

For example, solve x² − 5x + 6 < 0:

例如，解 x² − 5x + 6 < 0：

(x − 2)(x − 3) < 0

The roots are x = 2 and x = 3. Because the graph of y = (x − 2)(x − 3) opens upward, the expression is negative between the roots:

根为 x = 2 和 x = 3。由于 y = (x − 2)(x − 3) 的图像开口向上，表达式在两个根之间为负：

2 < x < 3

For x² − 5x + 6 > 0, the solution is the outside region:

对于 x² − 5x + 6 > 0，解集为外部区域：

x < 2 or x > 3

Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. This is especially relevant when dealing with expressions such as (x − 1)(x + 2) ≥ 0.

请记住：不等式两边同时乘以或除以负数时，不等号方向必须反转。这一点在解形如 (x − 1)(x + 2) ≥ 0 的问题时尤其重要。

9. Quadratic Simultaneous Equations | 二次联立方程

In CIE A-Level Mathematics, you often solve a linear equation and a quadratic equation simultaneously. The standard method is substitution.

在 CIE A-Level 数学中，经常需要联立一个线性方程和一个二次方程进行求解。标准方法是代入法。

Example: solve y = 2x + 1 and y = x² + x + 1.

例：解方程组 y = 2x + 1 和 y = x² + x + 1。

Substitute the linear expression into the quadratic:

将线性表达式代入二次方程：

2x + 1 = x² + x + 1 → x² − x = 0 → x(x − 1) = 0

So x = 0 or x = 1. Using y = 2x + 1 gives y = 1 when x = 0, and y = 3 when x = 1.

所以 x = 0 或 x = 1。利用 y = 2x + 1，当 x = 0 时 y = 1；当 x = 1 时 y = 3。

After substitution, the resulting quadratic may have two, one, or zero real solutions. This corresponds to the line intersecting the parabola twice, touching it once, or not intersecting at all.

代入后得到的二次方程可能有两个、一个或零个实数解。这分别对应直线与抛物线交于两点、相切于一点或完全没有交点。

10. Transformations of Quadratic Functions | 二次函数的变换

You should be able to relate transformations of the graph y = f(x) to changes in its equation.

你需要能够将 y = f(x) 的图像变换与方程的变化联系起来。

For a quadratic function, the completed square form y = a(x − h)² + k makes transformations easy to identify.

对于二次函数，完全平方形式 y = a(x − h)² + k 使变换很容易识别。

y = (x − 2)² and y = x²: translation by 2 units in the positive x-direction.
y = (x − 2)² 与 y = x²：沿 x 轴正方向平移 2 个单位。
y = x² + 3: translation by 3 units in the positive y-direction.
y = x² + 3：沿 y 轴正方向平移 3 个单位。
y = −x²: reflection in the x-axis, changing a minimum into a maximum.
y = −x²：关于 x 轴反射，将最小值点变为最大值点。
y = 2x²: vertical stretch with scale factor 2.
y = 2x²：纵向拉伸，比例为 2。

When applying transformations to x-intercepts or turning points, apply the same operation to the key points of the original graph.

对 x 轴交点或顶点进行变换时，只需把同样的操作施加到原图像的关键点上。

11. Using the Discriminant to Find Tangents and Intersections | 用判别式求切线与交点

When a line y = mx + c intersects a quadratic curve y = ax² + bx + d, substitute the line into the curve to form a quadratic equation in x.Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

A-Level Mathematics: Quadratic Equations and Functions | A-Level 数学：二次方程与函数