Solving Quadratic Equations and Functions | 解二次方程与函数

📚 Solving Quadratic Equations and Functions | 解二次方程与函数

Quadratic equations and functions are one of the most important topics in IGCSE Mathematics. Understanding how to solve them and interpret their graphs is essential for exam success in Edexcel.

二次方程与函数是 IGCSE 数学中最重要的主题之一。在 Edexcel 考试中，理解如何解二次方程并解释其图像，是取得高分的关键。

1. What Is a Quadratic Equation? | 什么是二次方程？

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.

二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。

The highest power of the variable x is 2, which is why it is called ‘quadratic’ from the Latin word ‘quadratum’ meaning square.

变量 x 的最高次数是 2，因此这类方程被称为“二次”方程，源于拉丁语 “quadratum”，意为“平方”。

2. Standard Form and Key Terms | 标准形式与关键术语

The standard form of a quadratic equation is ax² + bx + c = 0. Each part has a name:

二次方程的标准形式为 ax² + bx + c = 0。每个部分都有名称：

ax² is the quadratic term or leading term.

ax² 是二次项或首项。
bx is the linear term.

bx 是一次项。
c is the constant term.

c 是常数项。
a is the coefficient of x² and cannot be zero.

a 是 x² 的系数，且不能为零。

In a quadratic function f(x) = ax² + bx + c, we often replace the left side by y when drawing a graph.

在二次函数 f(x) = ax² + bx + c 中，画图时我们常把左边记为 y。

3. Solving by Factorisation | 用因式分解法求解

When a quadratic equation can be factorised, we can solve it using the zero product property: if AB = 0, then A = 0 or B = 0.

当二次方程可以因式分解时，我们可以利用零积性质求解：若 AB = 0，则 A = 0 或 B = 0。

For example, solve x² – 5x + 6 = 0. First factorise: (x – 2)(x – 3) = 0. Then set each bracket to zero: x – 2 = 0 or x – 3 = 0. So x = 2 or x = 3.

例如，解 x² – 5x + 6 = 0。首先因式分解：(x – 2)(x – 3) = 0。然后令每个括号为零：x – 2 = 0 或 x – 3 = 0。所以 x = 2 或 x = 3。

(x – 2)(x – 3) = 0 ⇒ x = 2 or x = 3

4. Solving by Completing the Square | 用配方法求解

Completing the square rewrites x² + bx + c in the form (x + p)² + q. This is useful when factorisation is not easy.

配方法将 x² + bx + c 改写为 (x + p)² + q 的形式。当因式分解困难时，这种方法非常有用。

For x² + 6x + 5 = 0, take half of 6, square it to get 9, then write: (x + 3)² – 9 + 5 = 0. This simplifies to (x + 3)² – 4 = 0.

对于 x² + 6x + 5 = 0，取 6 的一半为 3，平方得 9，然后写成：(x + 3)² – 9 + 5 = 0。化简得 (x + 3)² – 4 = 0。

(x + 3)² = 4 ⇒ x + 3 = ±2 ⇒ x = -1 or x = -5

Remember the formula: x² + bx = (x + b/2)² – (b/2)².

记住公式：x² + bx = (x + b/2)² – (b/2)²。

5. The Quadratic Formula | 二次公式

The quadratic formula works for every quadratic equation. It is derived from completing the square, and it is given as:

二次公式适用于所有二次方程。它由配方法推导而来，公式如下：

x = (-b ± √(b² – 4ac)) / 2a

To use it, identify a, b and c from ax² + bx + c = 0, then substitute into the formula.

使用时，从 ax² + bx + c = 0 中识别 a、b、c，然后代入公式。

For example, solve 2x² + 3x – 2 = 0. Here a = 2, b = 3, c = -2. Substituting gives x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4, so x = 0.5 or x = -2.

例如，解 2x² + 3x – 2 = 0。这里 a = 2，b = 3，c = -2。代入得 x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4，因此 x = 0.5 或 x = -2。

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

The part under the square root in the quadratic formula, b² – 4ac, is called the discriminant.

二次公式中根号下的部分 b² – 4ac 称为判别式。

The discriminant tells us how many real roots a quadratic equation has:

判别式告诉我们二次方程有多少个实数根：

Discriminant value	Number of real roots	判别式值	实数根个数
b² – 4ac > 0	Two distinct roots	b² – 4ac > 0	两个不同实数根
b² – 4ac = 0	One repeated root	b² – 4ac = 0	一个重根
b² – 4ac < 0	No real roots	b² – 4ac < 0	没有实数根

In the Edexcel IGCSE exam, you may be asked to find the value of k for which an equation has exactly one root by setting the discriminant to zero.

在 Edexcel IGCSE 考试中，可能会要求你通过令判别式为零，求出使方程恰好有一个根时的 k 值。

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

The graph of a quadratic function y = ax² + bx + c is a parabola.

二次函数 y = ax² + bx + c 的图像是一条抛物线。

If a > 0, the parabola opens upwards with a minimum point.

若 a > 0，抛物线开口向上，有最低点。
If a < 0, the parabola opens downwards with a maximum point.

若 a < 0，抛物线开口向下，有最高点。

The turning point (minimum or maximum) is also called the vertex of the parabola.

最高点或最低点也称为抛物线的顶点。

y-intercept: c (where x = 0)

The y-intercept of the graph is simply the constant term c, because when x = 0, y = c.

图像的 y 截距就是常数项 c，因为当 x = 0 时，y = c。

8. Roots and the x-axis | 根与 x 轴

The roots of a quadratic equation ax² + bx + c = 0 are the x-coordinates of the points where the graph of y = ax² + bx + c crosses the x-axis.

二次方程 ax² + bx + c = 0 的根，是 y = ax² + bx + c 的图像与 x 轴交点的 x 坐标。

By looking at the graph, you can estimate the roots, or you can plot the graph to solve an equation approximately.

通过观察图形，你可以估算根的值；或者画图来近似解方程。

If the discriminant is negative, the graph does not touch the x-axis at all. If the discriminant is zero, the graph touches the x-axis at exactly one point, which is the vertex.

如果判别式为负，图像完全不接触 x 轴。如果判别式为零，图像与 x 轴恰好相切于一点，这一点就是顶点。

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

The axis of symmetry of a parabola is the vertical line that passes through the vertex.

抛物线的对称轴是经过顶点的竖直直线。

x = -b / 2a

This value is also the x-coordinate of the vertex. To find the y-coordinate, substitute this x-value into the original function.

这个值也是顶点的 x 坐标。要求 y 坐标，只需将这个 x 值代入原函数。

For y = 2x² – 8x + 3, the axis of symmetry is x = 8 / 4 = 2. Then y = 2(2)² – 8(2) + 3 = -5, so the vertex is (2, -5).

对于 y = 2x² – 8x + 3，对称轴为 x = 8 / 4 = 2。然后 y = 2(2)² – 8(2) + 3 = -5，所以顶点为 (2, -5)。

10. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities are solved by first finding the roots, then testing intervals on a number line or using a sketch of the graph.

解二次不等式，首先求根，然后在数轴上检验区间，或者利用图像草图。

For example, solve x² + x – 6 > 0. Factorise: (x + 3)(x – 2) > 0. The roots are x = -3 and x = 2.

例如，解 x² + x – 6 > 0。因式分解：(x + 3)(x – 2) > 0。根为 x = -3 和 x = 2。

x < -3 or x > 2

Because the quadratic opens upwards, the graph is above the x-axis outside the interval between the roots.

因为二次函数开口向上，所以在两根之外的区间，图像位于 x 轴上方。

11. Applications and Word Problems | 应用与文字题

Quadratic equations appear in many real-life situations, including projectile motion, area problems and number puzzles.

二次方程出现在许多现实情境中，例如抛体运动、面积问题和数字谜题。

For example, a rectangle has length x + 3 and width x. Its area is 40. Then x(x + 3) = 40, so x² + 3x – 40 = 0. Factorise: (x + 8)(x – 5) = 0. Since length cannot be negative, x = 5.

例如，一个长方形的长是 x + 3，宽是 x，面积为 40。则 x(x + 3) = 40，即 x² + 3x – 40 = 0。因式分解得 (x + 8)(x – 5) = 0。由于长度不能为负，所以 x = 5。

Always check whether your solutions make sense in the context of the problem. Discard negative or impossible answers when necessary.

始终检查你的解是否符合实际情境。必要时舍去负数或不可能的答案。

12. Common Exam Pitfalls and Summary | 常见考试陷阱与总结

Here are some common mistakes that students make in Edexcel IGCSE exams:

以下是学生在 Edexcel IGCSE 考试中常犯的一些错误：

Forgetting to set the equation to zero before factorising or using the formula.

在因式分解或使用公式前，忘记将方程化为等于零的形式。
Mixing up the signs of a, b and c when substituting into the quadratic formula.

代入二次公式时，弄错 a、b、c 的符号。
Giving only one root when both roots are valid.

当两个根都有效时，只写出一个根。
Not simplifying surds such as √12 to 2√3 in exact-answer questions.

在要求精确答案的题中，没有将 √12 化简为 2√3。

To succeed, practise factorisation, memorise the quadratic formula, and always draw a quick sketch of the parabola when solving inequalities or interpreting graphs.

要想取得好成绩，请练习因式分解，牢记二次公式，并在解不等式或解释图像时画一个快速草图。

With consistent practice, quadratic equations and functions become straightforward and even enjoyable. Good luck with your revision!

坚持练习，二次方程与函数就会变得简单而有趣。祝你复习顺利！

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Solving Quadratic Equations and Functions | 解二次方程与函数