📚 Quadratic Equations: Solving and Applications | 二次方程：求解与应用
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to measurement and word problems. This article brings together the standard form, three algebraic solution methods, the discriminant, and graphical interpretation, with exam-style examples to help you revise efficiently.
二次方程贯穿 IGCSE 数学大纲，从代数、图像到测量和应用题都会涉及。本文汇总标准形式、三种代数解法、判别式以及图像解释，并配有考试风格例题，帮助你高效复习。
1. What is a Quadratic Equation? | 什么是二次方程？
e constants and a ≠ 0. The highest power of the variable x is 2, which is why it is called quadratic.
一元二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数且 a ≠ 0。未知数 x 的最高次数是 2，因此称为二次方程。
ratic equations. However, x³ + x² - 1 = 0 is not quadratic because the highest power is 3.
例如，x² - 3x + 2 = 0 和 2x² + 5x - 7 = 0 都是二次方程。但 x³ + x² - 1 = 0 不是二次方程，因为最高次数是 3。
2. Standard Form and Coefficients | 标准形式与系数
d c are called the quadratic coefficient, linear coefficient and constant term respectively.
求解前，通常最好先把方程整理成标准形式 ax² + bx + c = 0。数 a、b、c 分别称为二次项系数、一次项系数和常数项。
² - 3x - 4 = 0 after multiplying by -1. Always check the sign of a when using the formula.
对于 3x - x² + 4 = 0，整理后得到 -x² + 3x + 4 = 0，也可以乘以 -1 写成 x² - 3x - 4 = 0。使用公式时务必检查 a 的符号。
3. Solving by Factorisation | 因式分解法
. You write ax² + bx + c = 0 as (px + q)(rx + s) = 0, then set each bracket equal to zero.
如果二次式容易因式分解，因式分解法通常最快。把 ax² + bx + c = 0 写成 (px + q)(rx + s) = 0，然后令每个括号等于零。
d add to give -5. The numbers are -2 and -3, so (x - 2)(x - 3) = 0, giving x = 2 or x = 3.
解方程 x² - 5x + 6 = 0。我们需要两个数，乘积为 6，和为 -5。这两个数是 -2 和 -3，因此 (x - 2)(x - 3) = 0，得到 x = 2 或 x = 3。
Not every quadratic factorises using integers, so you must be ready to use completing the square or the quadratic formula.
并非每个二次式都能用整数因式分解，所以你必须准备好使用配方法或求根公式。
4. Solving by Completing the Square | 配方法
Completing the square rewrites x² + bx + c as (x + p)² + q. This form is useful for solving, finding the vertex of a parabola, and proving the quadratic formula.
配方法把 x² + bx + c 写成 (x + p)² + q 的形式。这种形式对求解、求抛物线的顶点以及证明求根公式都很有用。
3)² - 4. Setting (x + 3)² - 4 = 0 gives (x + 3)² = 4, so x + 3 = ±2 and x = -1 or x = -5.
对于 x² + 6x + 5 = 0，6 的一半是 3，且 3² = 9，所以 x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4。令 (x + 3)² - 4 = 0，得到 (x + 3)² = 4，因此 x + 3 = ±2，x = -1 或 x = -5。
x² + bx = (x + b/2)² − (b/2)²
This identity is the key step in completing the square and also leads directly to the quadratic formula.
这个恒等式是配方的关键步骤，也直接导出求根公式。
5. The Quadratic Formula | 求根公式
When
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