📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from basic factorising to sketching parabolas and using the discriminant. This revision guide walks you through the key methods and graph skills you need for both Core and Extended papers.
二次方程贯穿 IGCSE 数学课程，从基础因式分解到绘制抛物线以及使用判别式。本复习指南将带你掌握核心卷和扩展卷所需的关键方法与图像技能。
1. What is a Quadratic Expression? | 什么是二次表达式？
A quadratic expression is an algebraic expression in which the highest power of the variable is 2. Its standard form is ax² + bx + c, where a, b and c are constants and a ≠ 0.
二次表达式是变量的最高次数为 2 的代数式。其标准形式为 ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。
x² + 7x are all quadratics. If a = 0, the expression becomes linear rather than quadratic.
例如，3x² + 5x − 2、x² − 9 和 −2x² + 7x 都是二次式。如果 a = 0，该式就变成一次式而不是二次式。
In IGCSE questions, you will see quadratics in expanding brackets, factorising, solving equations, plotting graphs and in modelling real-life problems such as projectile motion.
在 IGCSE 考题中，二次式会出现在展开括号、因式分解、解方程、绘制图像以及抛体运动等实际建模问题中。
- Quadratic term: ax² | 二次项：ax²
- Linear term: bx | 一次项：bx
- Constant term: c | 常数项：c
2. Expanding and Factorising Quadratics | 二次式的展开与因式分解
To expand two binomials such as (x + p)(x + q), multiply each term in the first bracket by each term in the second bracket. The result is x² + (p + q)x + pq.
要展开 (x + p)(x + q) 这样的两个二项式，需要将第一个括号中的每一项分别乘以第二个括号中的每一项。结果是 x² + (p + q)x + pq。
(x + p)(x + q) = x² + (p + q)x + pq
For example, expanding (x + 2)(x − 5) gives x² − 5x + 2x − 10, which simplifies to x² − 3x − 10.
例如，展开 (x + 2)(x − 5) 得到 x² − 5x + 2x − 10，化简后为 x² − 3x − 10。
Factorising is the reverse process. When the coefficient of x² is 1, you need to find two numbers that multiply to give c and add to give b in x² + bx + c.
因式分解是相反的过程。当 x² 的系数为 1 时，你需要找到两个数，使它们的乘积等于 x² + bx + c 中的 c，并且它们的和等于 b。
- Start with x² + bx + c | 从 x² + bx + c 开始
- Find two numbers with product c and sum b | 找到乘积为 c、和为 b 的两个数
- Write the answer as (x + p)(x + q) | 将答案写成 (x + p)(x + q)
3. Solving by Factorising | 因式分解法解二次方程
To solve a quadratic equation by factorising, first set the equation equal to zero. Then factorise the quadratic expression and use the zero product property: if two factors multiply to zero, at least one factor must be zero.
用因式分解法解二次方程时，首先令方程等于零。然后对二次表达式进行因式分解，并使用零乘积性质：如果两个因式的乘积为零，那么至少有一个因式为零。
ing gives (x − 3)(x − 4) = 0, so x − 3 = 0 or x − 4 = 0. The solutions are x = 3 or x = 4.
例如，解方程 x² − 7x + 12 = 0。因式分解得到 (x − 3)(x − 4) = 0，所以 x − 3 = 0 或 x − 4 = 0。解为 x = 3 或 x = 4。
to zero. Always rearrange the equation into the form ax² + bx + c = 0 before factorising.
一个常见错误是没有先将方程化为等于零的形式就进行因式分解。因式分解前，一定要先把方程整理成 ax² + bx + c = 0 的形式。
- Set the equation to zero | 令方程等于零
- Factorise the quadratic expression | 因式分解二次表达式
- Set each bracket equal to zero and solve | 令每个括号等于零并求解
4. Completing the Square | 配方法
Completing the square is a method of rewriting a quadratic expression in the form (x + p)² + q. This is useful for solving equations, finding the vertex of a graph and deriving the quadratic formula.
配方法是将二次表达式改写为 (x + p)² + q 的形式。这种方法在解方程、求图像顶点以及推导求根公式时都非常有用。
x² + bx + c = (x + b/2)² − (b/2)² + c
For example, x² + 8x + 7 can be written as (x + 4)² − 16 + 7, which simplifies to (x + 4)² − 9.
例如，x² + 8x + 7 可以写成 (x + 4)² − 16 + 7，化简后为 (x + 4)² − 9。
Once the expression is in completed square form, you can solve the equation by isolating the squared bracket and taking square roots.
将表达式写成完全平方形式后，就可以通过移项并开平方来解方程。
- Take half of the coefficient of x | 取 x 系数的一半
- Square it and subtract it | 将其平方并减去
- Add the original constant term | 再加上原来的常数项
5. The Quadratic Formula | 求根公式
especially useful when factorising is difficult or when the coefficients are not integers.
求根公式可以直接给出 ax² + bx + c = 0 的解。当因式分解比较困难或系数不是整数时，它尤其有用。
x = (−b ± √b² − 4ac) ÷ (2a)
olve 2x² + 3x − 2 = 0, use a = 2, b = 3 and c = −2. Substitute into the formula: x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4. This gives x = 1/2 or x = −2.
例如，解方程 2x² + 3x − 2 = 0，取 a = 2、b = 3、c = −2。代入公式：x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4。由此得到 x = 1/2 或 x = −2。
Remember to use brackets around the numerator when calculating, especially if b is negative. The formula works for all quadratic equations with real or complex coefficients, but at IGCSE level you will only need real solutions.
计算时记住给分子加上括号，尤其是当 b 为负数时。该公式适用于所有二次方程，但 IGCSE 阶段你只需要处理实数解。
- Identify a, b and c from ax² + bx + c = 0 | 从 ax² + bx + c = 0 中确定 a、b、c
- Substitute into the formula carefully | 仔细代入公式
- Simplify using correct order of operations | 按照正确运算顺序化简
6. The Discriminant | 判别式
ac. It tells you how many real roots a quadratic equation has without actually solving it.
判别式是求根公式中平方根号下的部分：D = b² − 4ac。它可以在不解方程的情况下告诉你二次方程有多少个实根。
| Discriminant | 判别式 | Number of real roots | 实根个数 |
|---|---|
| D > 0 | two distinct real roots | 两个不同实根 |
| D = 0 | one repeated real root | 一个重根 |
| D < 0 | no real roots | 没有实根 |
nant D = 2² − 4(1)(5) = 4 − 20 = −16. Since D is negative, the equation has no real roots.
例如，x² + 2x + 5 = 0 的判别式为 D = 2² − 4(1)(5) = 4 − 20 = −16。由于 D 为负数，该方程没有实根。
Exam questions often ask you to find the value of k for which a quadratic has equal roots. In that case, set the discriminant equal to zero and solve for k.
考试题经常要求你求出使二次方程有两个相等实根的 k 值。此时，令判别式等于零并解出 k。
7. Graphs of Quadratic Functions | 二次函数的图像
ola. If a is positive, the parabola opens upward, and if a is negative, it opens downward.
y = ax² + bx + c 的图像是一条抛物线。如果 a 为正，抛物线开口向上；如果 a 为负，抛物线开口向下。
= c. The x-intercepts, if they exist, are the real roots of the equation ax² + bx + c = 0.
图像与 y 轴的交点是常数项 c，因为当 x = 0 时，y = c。如果存在 x 轴交点，它们就是方程 ax² + bx + c = 0 的实根。
. Its equation is x = −b/(2a). This line passes through the turning point of the parabola.
抛物线有一条竖直对称轴，其方程为 x = −b/(2a)。这条对称轴经过抛物线的顶点。
- a > 0: parabola opens upward | a > 0：抛物线开口向上
- a < 0: parabola opens downward | a < 0：抛物线开口向下
- c: y-intercept | c：y 轴截距
8. Turning Points and Symmetry | 顶点与对称性
adratic is written in completed square form y = (x − h)² + k, the turning point is (h, k).
二次图像顶点的位置是曲线改变方向的地方。如果二次式写成完全平方形式 y = (x − h)² + k，那么顶点就是 (h, k)。
tten as y = (x − 3)² − 4. The turning point is (3, −4), and the axis of symmetry is x = 3.
例如，y = x² − 6x + 5 可以写成 y = (x − 3)² − 4。其顶点为 (3, −4)，对称轴为 x = 3。
If a is positive, the turning point is a minimum point. If a is negative, the turning point is a maximum point.
如果 a 为正，顶点是最低点；如果 a 为负，顶点是最高点。
ating x = −b/(2a) and then substituting this x-value into the original equation to find y.
你也可以先计算 x = −b/(2a)，再将这个 x 值代入原方程求出 y，从而找到顶点。
9. Using Graphs to Solve Equations | 利用图像解方程
+ 3 crosses the x-axis at x = 1 and x = 3, so these are the solutions to x² − 4x + 3 = 0.
二次函数图像可以通过读取交点的 x 坐标来解方程。例如，y = x² − 4x + 3 的图像与 x 轴交于 x = 1 和 x = 3，因此这些就是 x² − 4x + 3 = 0 的解。
axes. The x-coordinates of the points where the line cuts the parabola are the solutions.
要用图像解 x² − 4x + 3 = 2 这样的方程，在同一坐标系中画出水平线 y = 2。直线与抛物线的交点的 x 坐标就是方程的解。
IGCSE exam papers often provide a quadratic graph and ask you to use it to solve related equations. Make sure you give x-values to the accuracy requested, usually from the graph grid.
IGCSE 试卷通常会给出一条二次曲线，并让你利用它解相关方程。确保你按照题目要求的精度写出 x 值，通常是从图像网格中读取。
10. Exam-style Tips | 考试技巧
When solving quadratic equations, always set the equation to zero before factorising. Show all steps clearly to gain method marks even if a small arithmetic error occurs.
解二次方程时，一定要先将方程化为等于零的形式再进行因式分解。清晰展示所有步骤，这样即使出现小的计算错误，也能获得方法分。
Use the discriminant to check the nature of roots before attempting to factorise. If D is not a perfect square, the roots will be irrational and factorising over integers will not work well.
在尝试因式分解之前，先使用判别式判断根的性质。如果 D 不是完全平方数，根将是无理数，用整数因式分解就不会顺利。
For graph questions, label the turning point, intercepts and axis of symmetry. When reading values from a graph, use a ruler and be careful with the scale.
在图像题中，要标出顶点、截距和对称轴。从图像上读值时，要使用直尺并注意坐标刻度。
- Set equations to zero before factorising | 因式分解前令方程等于零
- Remember the ± sign in the quadratic formula | 记住求根公式中的 ± 号
- Check the sign of a to determine opening direction | 检查 a 的符号以判断开口方向
- Use completing the square for vertex and symmetry | 用配方法求顶点和对称轴
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