📚 Quadratic Equations: Factorising, Completing the Square, and the Quadratic Formula | IGCSE 数学：二次方程求解方法
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from basic algebraic manipulation to graph sketching and problem solving. This revision guide explains three core methods for solving quadratic equations: factorising, completing the square, and using the quadratic formula. You will also learn how the discriminant helps you predict the nature of the roots.
二次方程贯穿 IGCSE 数学课程的始终，从基础代数运算到函数图像草图绘制以及实际问题解决都会涉及。本复习指南讲解求解二次方程的三种核心方法：因式分解法、配方法和求根公式法。你还会学习如何利用判别式来判断根的性质。
1. What is a Quadratic Equation? | 什么是二次方程？
are constants and a ≠ 0. The highest power of the unknown x is 2, which gives the equation its name 'quadratic'.
二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。未知数 x 的最高次数是 2，因此这类方程被称为“二次方程”。
If a were equal to 0, the x² term would disappear and the equation would become linear, not quadratic. The condition a ≠ 0 is therefore essential.
如果 a = 0，x² 项就会消失，方程就变成一次方程而不再是二次方程。因此，a ≠ 0 这一条件非常重要。
2. Standard Form and Key Terms | 标准形式与关键术语
b and c are called coefficients: a is the coefficient of x², b is the coefficient of x, and c is the constant term.
二次方程的标准形式为 ax² + bx + c = 0。常数 a、b、c 称为系数：a 是 x² 的系数，b 是 x 的系数，c 是常数项。
A quadratic equation can have two real roots, one repeated real root, or no real roots. The roots are the values of x that satisfy the equation, also called solutions or x-intercepts on a graph.
二次方程可能有两个实数根、一个重复实数根或没有实数根。根就是满足方程的 x 值，也叫做解或图像上的 x 轴截距。
3. Solving by Factorising | 因式分解法
be written as a product of two linear brackets. For example, x² + 5x + 6 = (x + 2)(x + 3).
当二次式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。例如，x² + 5x + 6 = (x + 2)(x + 3)。
p)(x + q) = 0, then either x + p = 0 or x + q = 0. Solve each linear equation separately.
因式分解后，使用零乘积性质：如果 (x + p)(x + q) = 0，那么 x + p = 0 或 x + q = 0。分别解出每一个一元一次方程即可。
Always make sure one side of the equation is 0 before factorising. If the equation is not equal to 0, rearrange it into standard form first.
在因式分解之前，务必确保方程的一边为 0。如果方程不满足等于 0 的条件，先将其整理成标准形式。
4. Worked Example: Factorising | 例题：因式分解
that multiply to 12 and add to 7. These are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4) = 0.
解方程 x² + 7x + 12 = 0。我们需要找到两个数，它们的乘积为 12，和为 7。这两个数是 3 和 4，所以 x² + 7x + 12 = (x + 3)(x + 4) = 0。
3 = 0 gives x = -3, and x + 4 = 0 gives x = -4. Hence the solutions are x = -3 or x = -4.
令每个括号等于零：x + 3 = 0 得 x = -3，x + 4 = 0 得 x = -4。因此解为 x = -3 或 x = -4。
When the coefficient of x² is not 1, for example 2x² + 7x + 3, you need to split the middle term or use trial and error to factorise into (2x + 1)(x + 3).
当 x² 的系数不是 1 时，例如 2x² + 7x + 3，需要使用拆中间项法或试误法将其因式分解为 (2x + 1)(x + 3)。
5. Completing the Square | 配方法
Completing the square transforms a quadratic expression into the form a(x + p)² + q. This method is especially useful when factorising is difficult or when you need the vertex of a parabola.
配方法将二次式转化为 a(x + p)² + q 的形式。当因式分解较困难，或需要求抛物线的顶点时，该方法尤其有用。
For x² + bx, the key step is to add and subtract (b/2)². This creates a perfect square:
对于 x² + bx，关键步骤是加上并减去 (b/2)²。这样可以构造完全平方：
x² + bx + (b/2)² = (x + b/2)²
To solve an equation by completing the square, isolate the x² and x terms, add the completing square term to both sides, then take square roots.
使用配方法解方程时，先将 x² 和 x 项移到一边，将配方的常数项加到等式两边，然后开平方求解。
6. Worked Example: Completing the Square | 例题：配方法
nstant: x² + 6x = -2. Half of 6 is 3, and 3² = 9, so add 9 to both sides: x² + 6x + 9 = 7.
用配方法解方程 x² + 6x + 2 = 0。移常数项：x² + 6x = -2。6 的一半是 3，3² = 9，所以等式两边同时加 9：x² + 6x + 9 = 7。
The left side is (x + 3)², so (x + 3)² = 7. Take the square root of both sides: x + 3 = ±√7. Therefore x = -3 ± √7.
左边是完全平方 (x + 3)²，所以 (x + 3)² = 7。两边开平方：x + 3 = ±√7。因此 x = -3 ± √7。
If the coefficient of x² is not 1, first factor it out from the x² and x terms before completing the square.
如果 x² 的系数不是 1，需要先将其从 x² 和 x 项中提出，再进行配方。
7. The Quadratic Formula | 求根公式
atic equation. The formula is derived from completing the square on the general quadratic.
求根公式可以求出任意二次方程 ax² + bx + c = 0 的解。该公式由一般二次方程的配方法推导而来。
x = (-b ± √b² - 4ac) / (2a)
Substitute the values of a, b and c carefully, paying attention to negative signs. Simplify the square root and the fraction if possible.
代入 a、b、c 的数值时要仔细，尤其注意负号。如果可能，化简根号和分数。
The quadratic formula works even when the roots are irrational or when factorising is not straightforward. It is a reliable method for all quadratic equations.
即使根是无理数或因式分解不直接适用，求根公式仍然有效。它是适用于所有二次方程的可靠方法。
8. Worked Example: Quadratic Formula | 例题：求根公式
he quadratic formula. Here a = 2, b = 3 and c = -2. Substitute into the formula: x = (-3 ± √3² - 4 × 2 × -2) / (2 × 2).
用求根公式解方程 2x² + 3x - 2 = 0。这里 a = 2，b = 3，c = -2。代入公式：x = (-3 ± √3² - 4 × 2 × -2) / (2 × 2)。
Simplify the discriminant: 3² - 4 × 2 × -2 = 9 + 16 = 25. So x = (-3 ± √25) / 4 = (-3 ± 5) / 4.
化简判别式：3² - 4 × 2 × -2 = 9 + 16 = 25。所以 x = (-3 ± √25) / 4 = (-3 ± 5) / 4。
5) / 4 = 2/4 = 1/2, and x = (-3 - 5) / 4 = -8/4 = -2. The solutions are x = 1/2 or x = -2.
得到两个值：x = (-3 + 5) / 4 = 2/4 = 1/2，以及 x = (-3 - 5) / 4 = -8/4 = -2。解为 x = 1/2 或 x = -2。
9. Discriminant and Nature of Roots | 判别式与根的性质
The expression b² - 4ac inside the square root of the quadratic formula is called the discriminant. It determines the nature of the roots without solving the equation.
求根公式根号内的表达式 b² - 4ac 称为判别式。它在不求解方程的情况下就能判断根的性质。
= 0, there is exactly one repeated real root. If b² - 4ac < 0, there are no real roots.
如果 b² - 4ac > 0，方程有两个不相等的实数根。如果 b² - 4ac = 0，方程有一个重复的实数根。如果 b² - 4ac < 0，方程没有实数根。
| Discriminant b² - 4ac | Nature of roots | Graph |
|---|---|---|
| > 0 | Two distinct real roots | Cuts x-axis at two points |
| = 0 | One repeated real root | Touches x-axis at one point |
| < 0 | No real roots | Does not meet x-axis |
10. Choosing the Best Method | 选择最佳方法
tally. This is usually the case when a = 1 and the roots are integers or simple fractions.
当二次式可以快速心算因式分解时，使用因式分解法。这通常适用于 a = 1 且根为整数或简单分数的情形。
Use completing the square when the question asks for the vertex form, maximum or minimum points, or when the equation is given in a form close to a perfect square.
当题目要求转化为顶点式、求最大值或最小值点，或方程形式接近完全平方时，使用配方法。
Use the quadratic formula when the coefficients are large, the roots are irrational, or factorising attempts do not work. It is the most general method.
当系数较大、根为无理数或因式分解难以进行时，使用求根公式。它是最通用的方法。
11. Common Mistakes | 常见错误
writing x² + 5x + 6 = 12 as (x + 2)(x + 3) = 12 does not allow the zero product property.
一个常见错误是在因式分解前忘记将方程化为等于
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图