📚 IGCSE Maths Unit 8: Quadratic Equations and Graphs | IGCSE 数学第8单元：二次方程与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graph sketching and real-world modelling. This unit brings together factorising, completing the square, the quadratic formula and the geometry of parabolas. The aim is to build reliable, exam-ready techniques rather than isolated rules.
二次方程贯穿 IGCSE 数学课程，从代数运算到函数图像再到实际问题建模都有涉及。本单元将因式分解、配方法、求根公式和抛物线几何性质整合在一起，帮助你形成可靠、适合考试的解题方法，而不是记忆零散规则。
1. What Is a Quadratic Equation? | 什么是二次方程？
a, b and c are constants and a ≠ 0. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.
一元二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a ≠ 0。ax² 叫作二次项，bx 叫作一次项，c 是常数项。
2. Standard Form and Coefficients | 标准形式与系数
nd c = -7. Identifying a, b and c correctly is essential for the formula and discriminant.
在选择解法之前，始终要把二次方程整理成 ax² + bx + c = 0 的形式。例如，3x - 2x² = 7 变形为 -2x² + 3x - 7 = 0，所以 a = -2，b = 3，c = -7。正确识别 a、b、c 对求根公式和判别式至关重要。
ax² + bx + c = 0, a ≠ 0
ax² + bx + c = 0，a ≠ 0
3. Solving by Factorising | 因式分解法
ping. For example, x² + 5x + 6 = 0 factors as (x + 2)(x + 3) = 0, giving x = -2 or x = -3.
当二次方程有简单的整数根时，因式分解法非常有效。找出两个数，它们的乘积是 ac，和是 b，然后拆项并分组分解。例如，x² + 5x + 6 = 0 分解为 (x + 2)(x + 3) = 0，得到 x = -2 或 x = -3。
Never cancel x from both sides unless you are certain that x ≠ 0, because doing so may remove a valid root. Always set the equation to zero before factorising.
不要随意在方程两边约去 x，除非你能确定 x ≠ 0，否则可能丢失一个有效的根。在因式分解之前，一定要先把方程化为等于零的形式。
4. Completing the Square | 配方法
6x + 5 = 0, write (x + 3)² - 4 = 0, then (x + 3)² = 4, so x + 3 = ±2 and x = -1 or x = -5.
配方法把 x² + bx + c 转化为 (x + p)² + q 的形式。这种方法能显示抛物线的顶点，并且在因式分解困难时仍然有效。对于 x² + 6x + 5 = 0，写成 (x + 3)² - 4 = 0，然后 (x + 3)² = 4，所以 x + 3 = ±2，得到 x = -1 或 x = -5。
x² + bx + c = (x + b/2)² - (b/2)² + c
x² + bx + c = (x + b/2)² - (b/2)² + c
5. The Quadratic Formula | 求根公式
ially useful when coefficients are large or decimals are involved. Substitute a, b and c carefully and simplify the square root step by step.
求根公式直接给出 ax² + bx + c = 0 的根。当系数较大或含有小数时尤其有用。代入 a、b、c 时要仔细，并逐步化简平方根。
x = (-b ± √b² - 4ac) / (2a)
x = (-b ± √b² - 4ac) / (2a)
For 2x² - 3x - 5 = 0, a = 2, b = -3, c = -5, so x = (3 ± √9 + 40) / 4 = (3 ± 7) / 4, giving x = 2.5 or x = -1.
对于 2x² - 3x - 5 = 0，a = 2，b = -3，c = -5，所以 x = (3 ± √9 + 40) / 4 = (3 ± 7) / 4，得到 x = 2.5 或 x = -1。
6. The Discriminant | 判别式
epeated real root. If Δ < 0, there are no real roots, although two complex roots exist.
判别式 Δ = b² - 4ac 可以告诉你二次方程有多少个实数根。如果 Δ > 0，有两个不同的实数根；如果 Δ = 0，有一个重复实数根；如果 Δ < 0，没有实数根，但存在两个复数根。
is a repeated root. In graph terms, the parabola touches the x-axis at exactly one point.
例如，x² - 4x + 4 = 0 的 Δ = 0，所以唯一的根是 x = 2，它是一个重复根。从图像上看，抛物线只在一点与 x 轴相切。
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. The y-intercept is c, and the x-intercepts are the real roots.
y = ax² + bx + c 的图像是抛物线。如果 a > 0，抛物线开口向上，有最低点；如果 a < 0，开口向下，有最高点。y 轴截距是 c，x 轴截距是实数根。
Sketching a quadratic graph requires the turning point, the axis of symmetry and the intercepts. You can find the roots by factorising, by the formula, or by reading them from the completed square form.
画二次函数图像需要确定转折点、对称轴和截距。你可以通过因式分解、求根公式或配方法后的形式来找到根。
8. Vertex and Axis of Symmetry | 顶点与对称轴
the x-coordinate, substitute it into the function to find the y-coordinate of the vertex.
顶点是抛物线的转折点。它的 x 坐标是 -b/(2a)，对称轴是竖直直线 x = -b/(2a)。求出 x 坐标后，将其代入函数，就可以得到顶点的 y 坐标。
x = -b / (2a), y = a(-b / (2a))² + b(-b / (2a)) + c
x = -b / (2a)，y = a(-b / (2a))² + b(-b / (2a)) + c
9. Applications to Word Problems | 实际问题应用
Many IGCSE problems ask you to form a quadratic equation from a context, such as area, projectile motion or profit. Define the unknown clearly, build the equation, solve it, and check whether both solutions make sense in the original situation.
许多 IGCSE 题目要求你根据实际情境列出二次方程，例如面积、抛体运动或利润问题。要清楚地设未知数，建立方程，求解，并检验两个解是否都符合原问题。
mes x² + 3x - 40 = 0. Factorising gives (x + 8)(x - 5) = 0, so x = 5 cm, rejecting x = -8.
一个长方形的长比宽多 3 cm，面积是 40 cm²。设宽为 x，则长为 x + 3，所以 x(x + 3) = 40，即 x² + 3x - 40 = 0。因式分解得 (x + 8)(x - 5) = 0，所以 x = 5 cm，舍弃 x = -8。
10. Exam Technique and Common Errors | 考试技巧与常见错误
xpression that may be zero, and always set the equation to zero before factorising. In graph questions, label the vertex and intercepts.
解题时要清晰展示所有步骤，尤其是在使用求根公式时。不要丢掉 ± 号，不要除以可能为零的表达式，在因式分解前一定要把方程化为标准形式并令其等于零。在图像题中，要标出顶点和截距。
- English: Check that a ≠ 0 before using the formula; if a = 0 the equation is linear.
中文：使用求根公式前检查 a ≠ 0；如果 a = 0 方程就是一次方程。 - English: Simplify surds fully: √48 = 4√3.
中文：完全化简根式：√48 = 4√3。 - English: Use exact values unless the question asks for decimals.
中文：除非题目要求小数，否则使用精确值。
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