Mastering Quadratic Equations | IGCSE二次方程完全指南

📚 Mastering Quadratic Equations | IGCSE二次方程完全指南

Quadratic equations are a cornerstone of IGCSE Mathematics. They appear in algebra, graphs, inequalities, and real-world problems. This guide gives you every essential method, with worked examples, to ensure you can solve them confidently.

二次方程是 IGCSE 数学的核心内容。它出现在代数、函数图像、不等式以及现实问题中。本指南将为你提供所有关键解法与典型例题，帮助你自信地解决各类二次方程问题。

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

• A quadratic equation can be written in the general form ax² + bx + c = 0, where a, b, c are constants and a ≠ 0.

  二次方程的一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

• The values of x that satisfy the equation are called its roots or solutions.

  满足方程的 x 值称为方程的根或解。

• Example: 2x² – 5x + 3 = 0 has a = 2, b = -5, c = 3.

  例如：2x² – 5x + 3 = 0 中 a = 2，b = -5，c = 3。

2. Factorising Quadratic Expressions | 因式分解二次式

• To factorise a quadratic of the form x² + bx + c, find two numbers that multiply to c and add to b.

  对于形如 x² + bx + c 的二次式，找到两个数，它们的乘积等于 c，和等于 b。

• For quadratics with a ≠ 1, find two numbers that multiply to ac and add to b, then rewrite the middle term and factor by grouping.

  当 a ≠ 1 时，找到两个数，它们的乘积等于 ac，和等于 b，然后改写中间项并分组因式分解。

• Example:

  例如：

  x² + 5x + 6 = (x + 2)(x + 3)

3. Solving by Factorisation | 用因式分解求解

• If (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0.

  若 (px + q)(rx + s) = 0，则必有 px + q = 0 或 rx + s = 0。

• Solve each linear equation to obtain the roots.

  分别解这两个一次方程即可得到方程的根。

• Example: x² – x – 6 = 0

  例如：x² – x – 6 = 0

  (x – 3)(x + 2) = 0 ⟹ x = 3 or x = -2

4. Completing the Square | 配方法

• Rearrange the equation into the form (x + p)² = q, then solve by taking the square root.

  将方程化为 (x + p)² = q 的形式，然后通过开平方求解。

• If a ≠ 1, divide the whole equation by a before completing the square.

  如果 a ≠ 1，先对等式两边同除以 a，再进行配方。

• Example: x² + 6x + 2 = 0

  例如：x² + 6x + 2 = 0

  (x + 3)² – 9 + 2 = 0 ⟹ (x + 3)² = 7 ⟹ x = -3 ± √7

5. The Quadratic Formula | 二次公式

• The roots of ax² + bx + c = 0 are given by the quadratic formula:

  方程 ax² + bx + c = 0 的根由二次公式给出：

  x = (-b ± √(b² – 4ac)) / (2a)

• This formula works for all quadratics, even those that do not factorise easily.

  该公式适用于所有二次方程，包括不容易因式分解的情况。

• Example: 2x² + 3x – 5 = 0

  例如：2x² + 3x – 5 = 0

  x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4 ⟹ x = 1 or x = -2.5

6. The Discriminant | 判别式

• The discriminant is defined as Δ = b² – 4ac.

  判别式定义为 Δ = b² – 4ac。

• If Δ > 0, the equation has two distinct real roots. If Δ = 0, it has one repeated real root. If Δ < 0, it has no real roots.

  若 Δ > 0，方程有两个不相等的实根；若 Δ = 0，方程有一个重根；若 Δ < 0，方程没有实根。

• Example: x² – 4x + 4 has Δ = 16 – 16 = 0, so it has one repeated root.

  例如：x² – 4x + 4 的 Δ = 16 – 16 = 0，所以它有一个重根。

  x² – 4x + 4 = 0 ⟹ (x – 2)² = 0 ⟹ x = 2

7. Quadratic Graphs | 二次函数图像

• The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upward; if a < 0, it opens downward.

  函数 y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时开口向上；当 a < 0 时开口向下。

• The axis of symmetry has equation x = -b / (2a), and the vertex lies on this line.

  对称轴的方程为 x = -b / (2a)，顶点位于这条直线上。

• The solutions of ax² + bx + c = 0 are the x-intercepts of the graph.

  方程 ax² + bx + c = 0 的解就是该图像与 x 轴交点的横坐标。

8. Solving Quadratic Inequalities | 解二次不等式

• To solve a quadratic inequality, first solve the corresponding equation, then test intervals on a number line.

  解二次不等式时，先解对应方程，再在数轴上检验各区间。

• The solution set may be a single interval or the union of two intervals.

  解集可能是一个区间，也可能是两个区间的并集。

• Example: solve x² – 3x + 2 > 0

  例如：解 x² – 3x + 2 > 0

  (x – 1)(x – 2) > 0 ⟹ x < 1 or x > 2

9. Word Problems with Quadratics | 二次方程应用题

• Quadratic equations often model areas, projectiles, profit, and other real-life situations.

  二次方程常用于求解面积、抛体运动、利润等现实问题。

• Set up the equation from the problem, solve it, and then check whether each root makes sense in the original context.

  根据问题列出方程，求解后检查每个根在原有情境下是否合理。

• Example: A rectangle has area 48 m² and its length is 2 m more than its width. Find the width.

  例如：一个长方形面积为 48 m²，长比宽多 2 m，求宽。

  x(x + 2) = 48 ⟹ x² + 2x – 48 = 0 ⟹ (x + 8)(x – 6) = 0 ⟹ x = 6 (since x > 0)

10. Common Mistakes and Tips | 常见错误与技巧

• Always rearrange the equation into the form ax² + bx + c = 0 before solving.

  求解前务必先将方程整理为 ax² + bx + c = 0 的形式。

• If a quadratic is not easy to factorise, use the quadratic formula or completing the square.

  如果二次式不易因式分解，请使用二次公式或配方法。

• Check your answers by substituting them back into the original equation.

  将求得的根代回原方程，检验答案是否正确。

• Remember that a negative root may be rejected in real-life problems if it has no physical meaning.

  在应用题中，若负根没有实际意义，应舍去。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）