📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
are constants and a ≠ 0. This article walks through the main methods, graphs and exam techniques you need.
二次方程贯穿 IGCSE 数学课程，从纯代数到实际建模都会出现。二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。本文梳理你需要掌握的主要方法、图像与考试技巧。
1. Standard Form and Key Terms | 标准形式与关键术语
condition a ≠ 0 is essential; if a = 0, the equation becomes linear rather than quadratic.
二次方程的标准形式是 ax² + bx + c = 0，其中 a、b 和 c 是实数，且 a ≠ 0。a ≠ 0 这个条件至关重要；如果 a = 0，方程就变成一次方程而不是二次方程。
The term ax² is called the quadratic term, bx is the linear term, and c is the constant term. The coefficient a controls the width and direction of the parabola, while c gives the y-intercept.
ax² 项称为二次项，bx 项称为一次项，c 称为常数项。系数 a 决定抛物线的宽度和开口方向，而 c 给出 y 轴截距。
ation into standard form before solving. For example, x² = 5x - 6 becomes x² - 5x + 6 = 0.
在 IGCSE 题目中，你通常需要先把方程整理成标准形式再求解。例如 x² = 5x - 6 要化为 x² - 5x + 6 = 0。
2. Solving by Factorising | 因式分解法
- 5x + 6 = 0, find two numbers whose product is 6 and whose sum is -5: they are -2 and -3.
当二次方程具有简单的整数根时，因式分解通常是最快的方法。要求解 x² - 5x + 6 = 0，需要找到两个数，它们的乘积为 6、和为 -5：这两个数是 -2 和 -3。
is zero, we set each bracket equal to zero: x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
因式分解后的形式是 (x - 2)(x - 3) = 0。由于乘积为零只有在至少一个因式为零时成立，我们令每个括号分别等于 0：x - 2 = 0 或 x - 3 = 0，解得 x = 2 或 x = 3。
erm or trial and error. The expression factorises to (2x + 1)(x + 3), so x = -½ or x = -3.
当 a ≠ 1 时，例如 2x² + 7x + 3 = 0，可以使用拆中间项或尝试法。该式分解为 (2x + 1)(x + 3)，所以 x = -½ 或 x = -3。
3. Completing the Square | 配方法
Completing the square transforms a quadratic into the form a(x - h)² + k. This is especially useful for finding the vertex and for deriving the quadratic formula.
配方法将二次式转化为 a(x - h)² + k 的形式。这对于求顶点以及推导求根公式特别有用。
ch is 3, square it to get 9, then add and subtract it: x² + 6x + 9 - 9 + 5 = (x + 3)² - 4.
对于 x² + 6x + 5，取 x 项系数的一半，即 3，将其平方得到 9，然后同时加上并减去它：x² + 6x + 9 - 9 + 5 = (x + 3)² - 4。
dd 4 to both sides: (x + 3)² = 4. Then take square roots: x + 3 = ±2, so x = -1 or x = -5.
要求解 (x + 3)² - 4 = 0，先把 4 移到等号右边：(x + 3)² = 4。然后开平方：x + 3 = ±2，因此 x = -1 或 x = -5。
k, the vertex is (h, k). This makes completing the square very useful for graph questions.
在配方法形式 y = (x - h)² + k 中，顶点就是 (h, k)。这使得配方法在图像题中非常有用。
4. The Quadratic Formula | 求根公式
be factorised easily. For ax² + bx + c = 0, the solutions are given by the formula below.
求根公式可以解任何二次方程，包括不容易因式分解的方程。对于 ax² + bx + c = 0，解由下面的公式给出。
x = (-b ± √b² - 4ac) / (2a)
e values of a, b and c before substituting. For 3x² + 5x - 2 = 0, a = 3, b = 5 and c = -2.
代入前一定要先写出 a、b、c 的值。对于 3x² + 5x - 2 = 0，a = 3，b = 5，c = -2。
Substituting gives x = (-5 ± √(5² - 4 × 3 × (-2))) / (2 × 3) = (-5 ± √49) / 6 = (-5 ± 7) / 6, so x = ⅓ or x = -2.
代入后得到 x = (-5 ± √(5² - 4 × 3 × (-2))) / (2 × 3) = (-5 ± √49) / 6 = (-5 ± 7) / 6，因此 x = ⅓ 或 x = -2。
Learn the formula exactly as it appears in the formula sheet, and practise using it with negative coefficients to avoid sign errors.
要准确记住公式表中的求根公式，并多练习含负系数的题目，以避免符号错误。
5. The Discriminant | 判别式
The discriminant is the expression b² - 4ac inside the square root of the quadratic formula. It tells you the number and type of roots without solving the equation.
判别式是求根公式中根号内的表达式 b² - 4ac。它可以在不求根的情况下告诉你根的数量和类型。
Δ = b² - 4ac
exactly one repeated real root. If Δ < 0, there are no real roots, but two complex roots.
如果 Δ > 0，方程有两个不同的实根。如果 Δ = 0，方程有一个重复的实根。如果 Δ < 0，方程没有实根，但有两个复数根。
xample, x² - 4x + 4 = 0 has Δ = (-4)² - 4 × 1 × 4 = 0, so it has one repeated root, x = 2.
例如 x² - 4x + 4 = 0 的判别式 Δ = (-4)² - 4 × 1 × 4 = 0，因此它有一个重复根 x = 2。
In graph terms, the discriminant tells you whether the parabola crosses the x-axis twice, touches it once, or does not meet it at all.
从图像角度看，判别式告诉你抛物线与 x 轴有两个交点、一个切点，还是完全没有交点。
6. 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。
g x = 0, giving the point (0, c). The x-intercepts are the real roots of ax² + bx + c = 0.
y 轴截距可通过令 x = 0 得到，即点 (0, c)。x 轴截距是方程 ax² + bx + c = 0 的实根。
Sketching a quadratic graph usually requires the vertex, the intercepts and the line of symmetry. Label these clearly on the diagram.
绘制二次函数图像通常需要顶点、截距和对称轴。在图上清楚地标注这些要素。
7. Vertex and Axis of Symmetry | 顶点与对称轴
x = -b / (2a), and the y-coordinate is found by substituting this value into the function.
顶点是抛物线的转折点。其 x 坐标由 x = -b / (2a) 给出，y 坐标通过将这个值代入函数求出。
x = -b / (2a)
a), which passes through the vertex and divides the parabola into two mirror-image halves.
对称轴是竖直线 x = -b / (2a)，它穿过顶点，将抛物线分成左右镜像对称的两半。
= (x - h)² + k, the vertex is simply (h, k). For y = (x - 3)² - 4, the vertex is (3, -4).
如果二次式是配方法形式 y = (x - h)² + k，顶点就是 (h, k)。对于 y = (x - 3)² - 4，顶点是 (3, -4)。
8. Roots and x-Intercepts | 根与 x 截距
are the x-coordinates where the graph crosses the x-axis. They are also called solutions or zeros.
方程 ax² + bx + c = 0 的根是图像与 x 轴交点的 x 坐标。它们也称为解或零点。
A parabola may cross the x-axis twice, touch it once, or not cross it at all. This corresponds exactly to the discriminant being positive, zero or negative.
抛物线可能与 x 轴相交两次、相切一次，或完全不相交。这恰好对应判别式为正、为零或为负的三种情况。
to the parabola at the vertex. For example, y = x² - 4x + 4 touches the x-axis at (2, 0).
当二次方程有重复根时，x 轴是抛物线在顶点处的切线。例如 y = x² - 4x + 4 在 (2, 0) 处与 x 轴相切。
9. Real-World Applications | 实际应用
Quadratic equations model many real situations, such as projectile motion, area optimisation, profit maximisation and stopping distances. The roots often represent physical events, such as when an object hits the ground.
二次方程可模拟许多实际情境，如抛体运动、面积优化、利润最大化和刹车距离。根通常表示物理事件，例如物体何时落地。
the time when the object reaches the ground. Negative time solutions are usually rejected in context.
对于高度 h = -5t² + 20t + 1 的抛体，令 h = 0 并求解可以得到物体到达地面的时间。在具体情境中，负时间解通常舍弃。
In optimisation, the maximum or minimum value occurs at the vertex. If a
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