📚 Quadratic Equations: Factoring, Formula and Graphs | IGCSE 数学二次方程：因式分解、公式法与图像
A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs and real-world modelling questions. Understanding how to solve quadratics by factorising, using the quadratic formula and completing the square gives you a reliable toolkit for exam success.
二次方程是IGCSE数学中最重要的主题之一。它出现在代数、图像和现实建模题中。掌握因式分解、公式法和配方法解二次方程，能为你提供可靠的考试成功工具。
1. What Is a Quadratic Equation? | 什么是二次方程？
iable is x². A typical form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
二次方程是一个次数为2的多项式方程。其变量的最高次幂是x²。典型形式为 ax² + bx + c = 0，其中a、b、c为常数且a ≠ 0。
tion becomes linear, not quadratic. Examples include x² - 5x + 6 = 0 and 2x² + 3x - 2 = 0.
如果a = 0，方程就变成一次方程，而不是二次方程。例子包括 x² - 5x + 6 = 0 和 2x² + 3x - 2 = 0。
2. Standard Form and Terminology | 标准形式与术语
coefficient a is called the leading coefficient, b is the linear coefficient, and c is the constant term.
标准形式为 ax² + bx + c = 0。系数a称为二次项系数，b称为一次项系数，c称为常数项。
andard form before solving. For example, x² + 4 = 5x should be written as x² - 5x + 4 = 0.
在求解之前，始终将方程整理为标准形式。例如 x² + 4 = 5x 应写成 x² - 5x + 4 = 0。
3. Solving by Factorising | 因式分解法
Factorising is usually the fastest method when the quadratic has simple integer roots. You look for two numbers whose product is ac and whose sum is b.
当二次方程有简单的整数根时，因式分解通常是最快的方法。你要寻找两个数，它们的乘积是ac，和是b。
mbers -2 and -3 multiply to 6 and add to -5. So (x - 2)(x - 3) = 0, giving x = 2 or x = 3.
对于 x² - 5x + 6 = 0，数 -2 和 -3 相乘得6、相加得 -5。因此 (x - 2)(x - 3) = 0，得到 x = 2 或 x = 3。
r the zero-product property: if AB = 0, then A = 0 or B = 0. This is why the method works.
记住零乘积性质：如果AB = 0，那么A = 0或B = 0。这就是该方法成立的原因。
4. Solving by the Quadratic Formula | 公式法
The quadratic formula works for any quadratic equation, even when factorising is difficult. It is derived from completing the square.
二次公式适用于任何二次方程，即使因式分解很困难。它由配方法推导而来。
x = [-b ± √b² - 4ac] / (2a)
Substitute a, b and c carefully. Simplify the square root and the fraction step by step to reduce sign errors.
仔细代入a、b和c。逐步化简平方根和分数，以减少符号错误。
For 2x² + 3x - 2 = 0, a = 2, b = 3, c = -2, so x = [-3 ± √9 + 16] / 4 = [-3 ± 5] / 4, giving x = 1/2 or x = -2.
对于 2x² + 3x - 2 = 0，a = 2，b = 3，c = -2，所以 x = [-3 ± √9 + 16] / 4 = [-3 ± 5] / 4，得到 x = 1/2 或 x = -2。
5. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful for solving, finding turning points and proving identities.
配方法将 ax² + bx + c 重写为 a(x + p)² + q 的形式。这对求解、求顶点和证明恒等式都很有用。
x² + 6x + 5 = (x + 3)² - 4
stant. For x² + 6x + 5, half of 6 is 3, and 3² = 9, so x² + 6x + 9 - 9 + 5 = (x + 3)² - 4.
取x系数的一半，平方，然后调整常数。对于x² + 6x + 5，6的一半是3，3² = 9，所以 x² + 6x + 9 - 9 + 5 = (x + 3)² - 4。
Setting (x + 3)² - 4 = 0 gives (x + 3)² = 4, so x + 3 = ±2, and x = -1 or x = -5.
令 (x + 3)² - 4 = 0，得到 (x + 3)² = 4，所以 x + 3 = ±2，因此 x = -1 或 x = -5。
6. The Discriminant | 判别式
in the quadratic formula: Δ = b² - 4ac. It tells you how many real roots the equation has.
判别式是二次公式中平方根内的表达式：Δ = b² - 4ac。它告诉你方程有多少个实数根。
ac = 0, there is exactly one repeated real root. If b² - 4ac < 0, there are no real roots.
如果 b² - 4ac > 0，有两个不同的实数根。如果 b² - 4ac = 0，有一个重复的实数根。如果 b² - 4ac < 0，没有实数根。
This is a powerful tool for questions that ask you to find the number of solutions without solving the equation fully.
对于要求你在不解完整方程的情况下求出解的个数的题目，这是一个强大的工具。
7. Graphs of Quadratic Functions | 二次函数图像
. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，有一个最低点。如果 a < 0，抛物线开口向下，有一个最高点。
intercept is at (0, c). The x-intercepts, or roots, are the solutions of ax² + bx + c = 0.
y轴截距在 (0, c)。x轴截距（即根）是 ax² + bx + c = 0 的解。
You can sketch a parabola by finding its roots, y-intercept, and turning point.
你可以通过求根、y轴截距和顶点来绘制抛物线草图。
8. Turning Point and Axis of Symmetry | 顶点与对称轴
The turning point is the highest or lowest point of the parabola. The axis of symmetry is a vertical line through the turning point.
顶点是抛物线的最高点或最低点。对称轴是通过顶点的竖直直线。
x = -b / (2a)
point is x = -b / (2a). Substitute this value into the equation to find the y-coordinate.
顶点的x坐标是 x = -b / (2a)。将这个值代入方程即可求出y坐标。
(x + p)² + q, the turning point is (-p, q). This is often quicker than using the formula.
在配方法形式 y = (x + p)² + q 中，顶点为 (-p, q)。这通常比使用公式更快。
try is x = -p. For y = (x + 3)² - 4, the turning point is (-3, -4) and the axis is x = -3.
对称轴是 x = -p。对于 y = (x + 3)² - 4，顶点是 (-3, -4)，对称轴是 x = -3。
9. Real-World Applications | 实际应用
Quadratic equations model many real situations, such as projectile motion, area problems, and profit maximisation. The roots often answer questions about when an object hits the ground or when a variable reaches zero.
二次方程可以模拟许多实际情况，如抛体运动、面积问题和利润最大化。根通常回答物体何时落地或变量何时达到零的问题。
own upwards might be h = -5t² + 20t + 1. Solving h = 0 gives the time when the ball lands.
例如，向上抛出的球的高度可能是 h = -5t² + 20t + 1。解 h = 0 可得到球落地的时间。
Remember that real-life solutions must make sense: reject negative times or impossible lengths.
记住现实生活中的解必须有意义：舍弃负的时间或不可能的长度。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is forgetting to set the equation to zero before factorising. Do not solve x² - 5x + 6 by guessing values without setting the expression equal to zero first.
一个常见错误是在因式分解前忘记将方程设为零。不要在没有先将表达式设为零的情况下猜 x² - 5x + 6 的解。
Another mistake is mishandling signs in the quadratic formula. Write brackets around negative b and around the entire discriminant to avoid errors.
另一个错误是在二次公式中错误处理符号。给 -b 和整个判别式加上括号，以避免错误。
Always check your answers by substituting them back into the original equation. This is quick and improves accuracy.
始终将答案代回原方程进行检验。这很快，并且能提高准确性。
In graphing questions, label the intercepts and turning point clearly. Marks are often awarded for these key features.
在图像题中，清楚标出截距和顶点。这些关键特征通常有分数可拿。
11. Practice Questions | 练习题
Try these IGCSE-style questions to test your understanding.
尝试以下IGCSE风格的题目来测试你的理解。
Solve by factorising: x² - 7x + 12 = 0. Answer: x = 3 or x = 4.
用因式分解法解：x² - 7x + 12 = 0。答案：x = 3 或 x = 4。
Solve using the quadratic formula: 3x² - 4x - 2 = 0. Answer: x = [4 ± √40] / 6 = [2 ± √10] / 3.
用公式法解：3x² - 4x - 2 = 0。答案：x = [4 ± √40] / 6 = [2 ± √10] / 3。
Find the turning point of y = x² - 4x + 1. Answer: (2, -3).
求 y = x² - 4x + 1 的顶点。答案：(2, -3)。
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