📚 Solving Quadratic Equations | 二次方程求解
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to real-life modelling. This revision guide covers the main methods for solving quadratics: factorising, completing the square, and the quadratic formula. It also explains the discriminant, graphical interpretation, and common exam pitfalls.
二次方程贯穿 IGCSE 数学考试大纲，从代数、图像到实际建模均有涉及。本篇复习指南涵盖求解二次方程的主要方法：因式分解法、配方法和二次公式法，同时解释判别式、图像意义以及常见考试失分点。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree 2. It contains a variable raised to the power of 2 as its highest power. The general shape of a quadratic graph is a parabola.
二次方程是一个次数为 2 的多项式方程，其中变量的最高次幂为 2。二次函数的图像通常是一条抛物线。
The standard form of a quadratic equation is:
二次方程的标准形式为：
ax² + bx + c = 0
a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.
其中 a、b、c 为常数，且 a ≠ 0。如果 a = 0，方程就变成一次方程，不再是二次方程。
For example, x² - 5x + 6 = 0 is quadratic with a = 1, b = -5, c = 6.
例如，x² - 5x + 6 = 0 是二次方程，其中 a = 1，b = -5，c = 6。
2. Standard Form and Coefficients | 标准形式与系数
. This makes the coefficients a, b and c clear and is essential for using the quadratic formula or the discriminant.
求解前，务必先把方程整理成标准形式 ax² + bx + c = 0。这样可以清楚看出系数 a、b、c，也是使用二次公式或判别式的前提。
For example, 3x = 2 - x² should be rearranged as:
例如，3x = 2 - x² 应整理为：
x² + 3x - 2 = 0
coefficient a must be positive in most exam answers, but any nonzero value is acceptable.
此时 a = 1，b = 3，c = -2。考试答案中通常把首项系数 a 写成正数，但只要 a 不为零即可。
- Always move all terms to one side so the other side is 0.
- 总是把所有项移到等号一边，使另一边为 0。
- Keep the x² term first, then the x term, then the constant.
- 按 x² 项、x 项、常数项的顺序排列。
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic has simple integer roots. If the product of two expressions is zero, then at least one of the expressions must be zero. This is called the zero product property.
当二次方程的根为简单整数时，因式分解通常是最快的方法。如果两个式子的乘积为零，那么其中至少一个式子必须为零，这称为零乘积性质。
To solve x² - 5x + 6 = 0, factorise it as:
求解 x² - 5x + 6 = 0，先因式分解：
(x - 2)(x - 3) = 0
Then set each bracket equal to zero:
然后令每个括号分别等于零：
x - 2 = 0 or x - 3 = 0
x = 2 or x = 3. Always check by substituting both values back into the original equation.
因此 x = 2 或 x = 3。最后务必把两个值代回原方程检验。
For non-monic quadratics such as 2x² + 7x + 3 = 0, you can factorise by grouping:
对于首项系数不为 1 的二次方程，如 2x² + 7x + 3 = 0，可用分组分解法：
2x² + 7x + 3 = 2x² + 6x + x + 3
= 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0
So x = -1/2 or x = -3.
所以 x = -½ 或 x = -3。
4. Solving by Completing the Square | 配方法
ethod is useful for deriving the quadratic formula and for finding the vertex of a parabola.
配方法把二次方程改写为 (x + p)² + q = 0 的形式。这种方法常用于推导二次公式以及求抛物线的顶点。
For x² + 6x + 5 = 0, first move the constant to the right-hand side:
对于 x² + 6x + 5 = 0，先把常数项移到右边：
x² + 6x = -5
Take half the coefficient of x, which is 3, and square it to get 9. Add 9 to both sides:
取 x 项系数的一半，即 3，将其平方得到 9。两边同时加上 9：
x² + 6x + 9 = -5 + 9
(x + 3)² = 4
Now take the square root of both sides, remembering both positive and negative roots:
两边开平方，记得取正负两个根：
x + 3 = ±2
Therefore x = -1 or x = -5.
因此 x = -1 或 x = -5。
ctorised easily. It is also used to write y = x² + 6x + 5 in vertex form y = (x + 3)² - 4.
当二次方程不容易因式分解时，配方法尤为重要。它也用于把 y = x² + 6x + 5 写成顶点式 y = (x + 3)² - 4。
5. The Quadratic Formula | 二次公式
ard form. It is derived by completing the square on the general equation ax² + bx + c = 0.
二次公式可以求解任何标准形式的二次方程。它是由一般方程 ax² + bx + c = 0 通过配方法推导出来的。
x = [-b ± √b² - 4ac] / 2a
+ bx + c = 0. Use brackets carefully when substituting negative values into a calculator.
这里的 a、b、c 是 ax² + bx + c = 0 中的系数。在计算器里代入负数时，务必使用括号。
For example, solve 3x² - 2x - 4 = 0. Here a = 3, b = -2, c = -4:
例如，求解 3x² - 2x - 4 = 0。这里 a = 3，b = -2，c = -4：
x = [2 ± √4 + 48] / 6 = [2 ± √52] / 6 = [2 ± 2√13] / 6
So x = (1 ± √13) / 3. This can be left in exact surd form unless the question asks for decimals.
所以 x =
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