Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are a central topic in IGCSE Mathematics. Understanding how to solve and interpret them is essential for success in both Paper 2 and Paper 4. This guide provides a comprehensive review of the concepts, techniques, and common pitfalls you need to master.

二次方程是 IGCSE 数学的核心内容。理解如何求解和解释二次方程，对 Paper 2 和 Paper 4 的成功都至关重要。本指南全面复习相关概念、技巧和常见易错点，帮助你扎实掌握。

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

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Its standard form is:

二次方程是次数为 2 的多项式方程，即变量的最高次数为 2。其标准形式为：

ax² + bx + c = 0

Here, a, b and c are real numbers, and a ≠ 0. If a = 0, the equation becomes linear.

其中 a、b 和 c 是实数，且 a ≠ 0。如果 a = 0，方程就变成一次方程。

The equation is called “quadratic” because the highest exponent is 2, and “quad” historically relates to squares.

方程被称为“二次”是因为最高指数为 2，而 “quad” 在历史上与正方形相关。
A quadratic equation may have two distinct real roots, one repeated real root, or no real roots, depending on the discriminant.

二次方程可能有两个不同的实数根、一个重根或无实数根，这取决于判别式。

2. Expanding and Factorising | 展开与因式分解

Before solving a quadratic equation, you often need to expand or factorise quadratic expressions. Expanding means removing brackets, while factorising means writing the expression as a product of two linear factors.

在求解二次方程之前，通常需要展开或因式分解二次表达式。展开是指去掉括号，因式分解是指将表达式写成两个一次因式的乘积。

For example, the expression (x + 3)(x − 2) expands to x² + x − 6.

例如，表达式 (x + 3)(x − 2) 展开后得到 x² + x − 6。

Conversely, x² + x − 6 can be factorised back to (x + 3)(x − 2).

反过来，x² + x − 6 可以因式分解为 (x + 3)(x − 2)。

The product of two binomials (ax + b)(cx + d) expands according to the distributive law.

两个二项式 (ax + b)(cx + d) 的乘积按分配律展开。
When factorising x² + bx + c, look for two integers whose sum is b and whose product is c.

因式分解 x² + bx + c 时，寻找两个整数，使其和为 b、积为 c。

(x + 3)(x − 2) = x² + x − 6

3. Solving by Factorisation | 因式分解法求解

The factorisation method uses the zero-product property: if the product of two expressions is zero, then at least one of the factors must be zero.

因式分解法利用零乘积性质：若两个表达式的乘积为零，则至少一个因子必须为零。

Example: Solve x² − 4x − 5 = 0.

示例：解方程 x² − 4x − 5 = 0。

Factorise: (x − 5)(x + 1) = 0

因式分解：(x − 5)(x + 1) = 0
Set each factor to zero: x − 5 = 0 or x + 1 = 0

令每个因子为零：x − 5 = 0 或 x + 1 = 0
Solve: x = 5 or x = −1

求解：x = 5 或 x = −1

Always check your solutions by substituting them back into the original equation.

务必通过代回原方程来检验解。

Not all quadratic expressions can be factorised easily using integers; in those cases, use the quadratic formula.

并非所有二次表达式都能用整数轻松因式分解；此时应使用求根公式。

4. The Quadratic Formula | 求根公式

The quadratic formula gives the roots of any quadratic equation ax² + bx + c = 0. It is derived from the method of completing the square and is valid for all values of a ≠ 0.

求根公式给出任意二次方程 ax² + bx + c = 0 的根。它由配方法推导而来，对所有 a ≠ 0 均适用。

x = (−b ± √(b² − 4ac)) / (2a)

To use the formula, identify a, b and c from the equation, substitute them into the formula, and simplify.

使用公式时，先识别方程中的 a、b 和 c，代入公式并化简。

Example: Solve 2x² + 3x − 2 = 0.

示例：解方程 2x² + 3x − 2 = 0。

Here a = 2, b = 3, c = −2. Then:

这里 a = 2，b = 3，c = −2。于是：

x = (−3 ± √(3² − 4×2×(−2))) / (2×2) = (−3 ± √25) / 4

So x = (−3 + 5)/4 = 0.5 or x = (−3 − 5)/4 = −2.

因此 x = (−3 + 5)/4 = 0.5 或 x = (−3 − 5)/4 = −2。

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

The expression b² − 4ac inside the quadratic formula is called the discriminant, often denoted by Δ.

求根公式中的 b² − 4ac 称为判别式，通常用 Δ 表示。

Discriminant (Δ)	Nature of roots	图形含义
Δ > 0	Two distinct real roots	抛物线交 x 轴于两个不同点
Δ = 0	One repeated real root	抛物线切 x 轴于一点（顶点）
Δ < 0	No real roots	抛物线不交 x 轴

It is important to distinguish between “real roots” and “no real roots” because the quadratic formula involves the square root of the discriminant; if Δ is negative, the square root is not a real number.

区分“实数根”和“无实数根”很重要，因为求根公式包含判别式的平方根；若 Δ 为负，则平方根不是实数。

In IGCSE, you are usually only asked to work with real roots and to state the number of roots.

在 IGCSE 中，通常只要求处理实数根，并说明根的个数。

6. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x − h)² + k. This form reveals the vertex of the parabola and is useful for proving the quadratic formula.

配方法将二次表达式写成 a(x − h)² + k 的形式。这种形式能显示抛物线的顶点，并用于推导求根公式。

Example: Complete the square for x² + 6x + 8.

示例：将 x² + 6x + 8 配方。

Take half of 6, square it: (6/2)² = 9. Then:

取 6 的一半，再平方：(6/2)² = 9。于是：

x² + 6x + 8 = (x + 3)² − 9 + 8 = (x + 3)² − 1

So the vertex of y = x² + 6x + 8 is at (−3, −1).

因此 y = x² + 6x + 8 的顶点为 (−3, −1)。

The general form is: x² + bx = (x + b/2)² − (b/2)².

一般形式为：x² + bx = (x + b/2)² − (b/2)²。
For an expression with a coefficient of x² that is not 1, factor out a first.

对于 x² 系数不为 1 的表达式，先提出 a。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of a quadratic function y = ax² + bx + c is a parabola. The sign of a determines the direction of the curve: if a > 0, it opens upwards; if a < 0, it opens downwards.

二次函数 y = ax² + bx + c 的图像是抛物线。a 的符号决定曲线的开口方向：若 a > 0，开口向上；若 a < 0，开口向下。

特征	公式/说明
对称轴 (axis of symmetry)	x = −b/(2a)
顶点 (vertex)	(−b/(2a), f(−b/(2a)))
y 截距	(0, c)
x 截距（根）	由求解 ax² + bx + c = 0 得到

If the discriminant is positive, there are two x-intercepts; if it is zero, the vertex touches the x-axis; if negative, there are no x-intercepts.

若判别式为正，则有两个 x 截距；若为零，则顶点接触 x 轴；若为负，则没有 x 截距。
You may be asked to sketch the graph, clearly labelling the vertex, intercepts, and axis of symmetry.

题目可能要求画示意图，并清晰标出顶点、截距和对称轴。

8. Vertex Form and Transformations | 顶点式与图像变换

The vertex form y = a(x − h)² + k makes it easy to read the vertex (h, k). It also shows how the graph is transformed from the basic parabola y = x².

顶点式 y = a(x − h)² + k 可以轻松读出顶点 (h, k)。它还显示图像如何从基本抛物线 y = x² 变换而来。

If a is positive and greater than 1, the parabola is stretched vertically; if between 0 and 1, it is compressed.

若 a 为正且大于 1，抛物线纵向拉长；若在 0 和 1 之间，则被压缩。
The term (x − h) shifts the graph horizontally: positive h moves it right, negative h moves it left.

项 (x − h) 使图像水平平移：正 h 向右移，负 h 向左移。
The constant k shifts the graph vertically: positive k moves it up, negative k moves it down.

常数 k 使图像垂直平移：正 k 向上移，负 k 向下移。

To convert from standard form to vertex form, use completing the square.

要将标准式化为顶点式，可以使用配方法。

9. Applications and Problem Solving | 应用与实际问题

Quadratic equations often model real-world situations such as projectile motion, area problems, and revenue optimization. You may need to form a quadratic equation from a word problem and then solve it.

二次方程常用来模拟现实情境，如抛体运动、面积问题和收益最优化。你可能需要从文字题中建立二次方程并求解。

Example: The area of a rectangle is 36 cm². Its length is 5 cm more than its width. Find the width.

示例：一个矩形的面积为 36 平方厘米，长比宽多 5 厘米，求宽。

Let the width be x cm. Then the length is (x + 5) cm. So x(x + 5) = 36, giving x² + 5x − 36 = 0.

设宽为 x 厘米，则长为 (x + 5) 厘米。于是 x(x + 5) = 36，即 x² + 5x − 36 = 0。

Factorise: (x + 9)(x − 4) = 0, so x = −9 (rejected) or x = 4. The width is 4 cm.

因式分解：(x + 9)(x − 4) = 0，所以 x = −9（舍去）或 x = 4。宽为 4 厘米。

Always interpret the solutions in the context of the problem and discard values that do not make sense (like negative lengths).

始终在问题情境中解读解，并舍去没有意义的解（如负长度）。
Show clearly how you form the equation and label your final answer with units.

清晰展示方程的建立过程，并在最终答案中注明单位。

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

Many students lose marks on quadratic equations due to avoidable mistakes. Here are some common pitfalls to avoid.

许多学生在二次方程上因可避免的错误而失分。以下是一些常见陷阱，需注意避免。

Forgetting to set the equation to zero before factorising or using the formula. Quadratic equations must be in the form ax² + bx + c = 0.

在因式分解或使用求根公式前，忘记将方程整理为零。二次方程必须化为 ax² + bx + c = 0 的形式。
Sign errors when substituting negative values into the quadratic formula. Always use brackets when substituting.

将负值代入求根公式时出现符号错误。代入时一定要加括号。
Confusing the direction of the parabola when a is negative. Remember that a < 0 opens downwards.

当 a 为负时，混淆抛物线开口方向。记住 a < 0 开口向下。
Not checking whether a solution is valid in the original equation or context (e.g., rejecting negative lengths).

未检查解在原方程或实际情境中是否有效（如舍去负长度）。

For Paper 2, you may need to solve quadratic equations using a calculator’s polynomial solver if allowed, but you must still know the algebraic methods for non-calculator questions.

在 Paper 2 中，如果允许使用计算器，你可能需要利用计算器的多项式求解功能，但在非计算器题目中仍必须掌握代数方法。

11. Practice Questions | 练习题

Test your understanding with these short questions. Solve each equation and sketch the corresponding graph where possible.

用以下短题测试你的理解。解每个方程，并在可能的情况下画出相应图像。

Solve x² − 7x + 10 = 0 by factorisation.

用因式分解法解 x² − 7x + 10 = 0。
Use the quadratic formula to solve 3x² + x − 2 = 0.

用求根公式解 3x² + x − 2 = 0。
Find the discriminant of 4x² − 4x + 1 = 0 and state the number of real roots.

求 4x² − 4x + 1 = 0 的判别式，并说明实数根的个数。
Complete the square for x² − 6x + 11 and write the vertex of y = x² − 6x + 11.

将 x² − 6x + 11 配方，并写出 y = x² − 6x + 11 的顶点。
A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the length of the other leg (use a quadratic equation).

一个直角三角形斜边为 13 厘米，一条直角边为 5 厘米。求另一条直角边的长度（用二次方程）。

Answers: 1) x = 2, 5; 2) x = 0.5 or x = −2; 3) Δ = 0, one repeated root; 4) (x − 3)² + 2, vertex (3, 2); 5) 12 cm (by Pythagoras: x² + 25 = 169).

答案：1) x = 2, 5；2) x = 0.5 或 x = −2；3) Δ = 0，一个重根；4) (x − 3)² + 2，顶点 (3, 2)；5) 12 厘米（根据勾股定理：x² + 25 = 169）。

12. Summary | 总结

In this review, we covered the standard form of a quadratic equation, factorisation, the quadratic formula, the discriminant, completing the square, graph sketching, transformations, real-world applications, and common pitfalls.

本复习涵盖了二次方程的标准形式、因式分解、求根公式、判别式、配方法、图像绘制、图像变换、实际应用和常见错误。

Remember to practise solving quadratic equations fluently using all methods, and always check your answers. With regular practice, you will approach any quadratic problem with confidence.

请记住流畅地使用所有方法练习解二次方程，并随时检查答案。通过定期练习，你将自信地应对任何二次方程问题。

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Mastering Quadratic Equations | 掌握二次方程