📚 Algebra 8: Quadratic Equations | 代数8：二次方程
Quadratic equations form a cornerstone of IGCSE algebra, extending your skills from linear equations to second-degree polynomials. In this chapter, you will learn how to recognise a quadratic, solve it by factorisation, apply the quadratic formula, complete the square, interpret graphs, and use the discriminant to analyse roots. Mastery of these techniques is essential for tackling higher-tier problems and laying the groundwork for advanced functions and calculus.
二次方程是 IGCSE 代数的基石，它将你的技能从一次方程扩展到二次多项式。在本章中，你将学习如何识别二次方程，通过因式分解法求解，运用求根公式，完成配方法，解读图像，并利用判别式分析根的性质。掌握这些技巧对于解决高阶问题以及为后续学习高级函数与微积分打下基础至关重要。
1. What Is a Quadratic Equation? | 什么是二次方程？
a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which produces a curved graph called a parabola. Quadratic equations may have two real solutions, one repeated real solution, or no real solutions, depending on the discriminant.
二次方程是指任何可以写成 ax² + bx + c = 0 形式的二次多项式方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数为 2，其图像是一条称为抛物线的曲线。根据判别式的不同，二次方程可能有两个实数解、一个重复实数解，或没有实数解。
2. Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0
0. Always ensure the coefficient of x² is positive when possible for easier factorisation.
在求解任何二次方程之前，必须将其整理成标准形式：所有项移到一边并等于零，x 的幂按降序排列。例如，3x - x² = 5 变为 -x² + 3x - 5 = 0，然后乘以 -1 得到 x² - 3x + 5 = 0。尽量确保 x² 的系数为正，以便于因式分解。
3. Solving by Factorising | 因式分解法求解
lways check by expanding back. Factorising works efficiently when the roots are integers or simple fractions, and it often appears in non-calculator exam papers.
当二次方程能够写成两个一次因式乘积时，令每个因式等于零即可求得解。对于 x² + 5x + 6 = 0，分解为 (x + 2)(x + 3) = 0，得到 x = -2 或 x = -3。务必通过展开验算。当根为整数或简单分数时，因式分解法非常高效，且经常出现在非计算器试卷中。
- Step 1: Write in standard form ax² + bx + c = 0.
- Step 2: Find two numbers that multiply to ac and add to b.
- Step 3: Split the middle term and factorise by grouping, or use trial factors directly.
- Step 4: Solve each bracket = 0.
- 步骤 1：写成标准形式 ax² + bx + c = 0。
- 步骤 2：找出相乘得 ac、相加得 b 的两个数。
- 步骤 3：拆解中间项并分组分解，或直接尝试因式。
- 步骤 4：解每个括号等于零。
4. The Quadratic Formula | 二次公式法
The quadratic formula solves any equation of the form ax² + bx + c = 0. The formula is:
x = (-b ± √b² - 4ac) / 2a
Simply substitute the values of a, b and c. Be careful with signs, and always calculate the discriminant b² - 4ac first to determine the nature of the roots. This method is reliable even when factorisation is impossible or involves irrational roots.
二次公式可以求解任何形如 ax² + bx + c = 0 的方程。公式为：
x = (-b ± √b² - 4ac) / 2a
只需代入 a、b、c 的值。注意符号，并且始终先计算判别式 b² - 4ac 以确定根的性质。即使当因式分解不可行或包含无理根时，这个方法依然可靠。
5. Completing the Square | 配方法
e to obtain x = -1 or x = -5. For coefficients of x² not equal to 1, factor out the leading coefficient first.
配方法将二次方程重写为 a(x + p)² + q = 0 的形式。它在推导求根公式、寻找抛物线顶点以及方程无法整洁因式分解时特别有用。对于 x² + 6x + 5 = 0，加减 (6/2)² = 9 得到 (x + 3)² - 4 = 0，然后解得 x = -1 或 x = -5。如果 x² 系数不为 1，需先提取首项系数。
6. Graphical Solutions | 图像解法
x-intercepts where the curve crosses the x-axis. A quadratic with two distinct real roots crosses the x-axis twice; a repeated root touches it at a single point (tangent); and no real roots means the graph stays entirely above or below the x-axis. You may be asked to estimate solutions from a given graph or sketch a graph to solve an equation.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解就是曲线与 x 轴交点的横坐标。有两个相异实根时，抛物线与 x 轴相交两次；重根时与 x 轴相切于一点；无实根时图像完全位于 x 轴上方或下方。考题可能要求根据给定图像估算解，或通过绘制草图来解方程。
7. The Discriminant | 判别式
The discriminant Δ = b² - 4ac determines the number and type of roots:
- If Δ > 0: two distinct real roots.
- If Δ = 0: one repeated real root (or two equal roots).
- If Δ < 0: no real roots (two complex roots, not required at IGCSE).
Δ < 0, or to find the value of k such that the equation has equal roots by setting Δ = 0.
判别式 Δ = b² - 4ac 决定了根的数量与类型：
- 若 Δ > 0：两个相异实根。
- 若 Δ = 0：一个重复实根（或两个相等实根）。
- 若 Δ < 0：没有实根（两个复数根，IGCSE 不作要求）。
许多考题要求你通过证明 Δ < 0 来说明一个二次方程无实根，或者通过令 Δ = 0 求出使得方程有等根的 k 值。
8. Word Problems Involving Quadratics | 二次方程应用题
Quadratic equations often model area, projectile motion, and number problems. For example: "The product of two consecutive positive integers is 72. Find the integers." riables, form the equation, solve, and interpret the context to reject extraneous solutions.
二次方程常用于面积、抛体运动和数字问题建模。例如：“两个连续正整数的乘积是 72，求这两个整数。”设第一个为 n，则下一个为 n+1，得 n(n+1)=72 → n²+n-72=0 → (n+9)(n-8)=0 → n=8（取正整数解）。解题时需设定变量、建立方程、求解，并结合题意舍去多余解。
9. Common Mistakes and Tips | 常见错误与技巧
Students often forget to set the equation to zero before factorising, misinterpret the signs when substituting into the quadratic formula, or incorrectly apply square roots in completing the square. To avoid these, always write STORM (Standard form, Two brackets, Order of roots, Remember to check, Meaning final). Practise with both calculator and non-calculator questions, and underline key values of a, b, c to reduce sign errors. Familiarise yourself with the formula sheet, but also memorise the quadratic formula for speed.
学生常犯的错误包括：因式分解前忘记将方程设为零，代入求根公式时搞错符号，或在配方时错误运用平方根。为避免这些错误，建议遵循“STORM”法则（标准形式、两个括号、根的顺序、记住检查、最终含义）。多练习计算器和非计算器题目，标出 a、b、c 的关键值以减少符号错误。熟悉公式表，但也要记住求根公式以提高解题速度。
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