📚 Quadratic Equations and Graphs: IGCSE Maths Core Revision | IGCSE 数学二次方程与图像核心考点
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and problem solving to graphs of parabolas and modelling real-world situations. This revision guide covers factorising, completing the square, the quadratic formula, the discriminant, graph features and exam-style applications.
二次方程贯穿 IGCSE 数学课程，从代数运算、实际问题建模到抛物线图像分析都离不开它。本文系统梳理因式分解、配方法、求根公式、判别式、图像特征以及考试常见应用题。
1. Definition and Standard Form | 定义与标准形式
A quadratic equation in one variable is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is essential: if a = 0, the equation becomes linear. The highest power of the variable x is 2, which gives the equation its characteristic U-shaped or n-shaped graph.
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 为常数且 a ≠ 0。a ≠ 0 的条件非常重要：若 a = 0，方程就退化为一次方程。未知量 x 的最高次数为 2，这使得它的图像具有典型的 U 形或倒 U 形。
For example, x² – 3x + 2 = 0, 2x² + 5x – 7 = 0 and 4x² – 9 = 0 are all quadratic equations. The last one can also be written as 4x² + 0x – 9 = 0, so b can be zero.
例如，x² – 3x + 2 = 0、2x² + 5x – 7 = 0 和 4x² – 9 = 0 都是二次方程。最后一个方程也可写成 4x² + 0x – 9 = 0，因此 b 可以为零。
2. Solving by Factorising | 因式分解法求解
Factorising is usually the fastest method when the quadratic has integer roots. Write ax² + bx + c as (px + q)(rx + s), expand to check, then use the zero-product rule: if a product equals zero, at least one factor must be zero.
因式分解通常是在方程有整数根时最快捷的方法。先将 ax² + bx + c 写成 (px + q)(rx + s)，再展开检查，然后利用零乘积法则：若乘积为零，则至少有一个因式为零。
Example: Solve x² – 5x + 6 = 0. Factorising gives (x – 2)(x – 3) = 0, so x – 2 = 0 or x – 3 = 0, hence x = 2 or x = 3.
例如，解 x² – 5x + 6 = 0。因式分解得 (x – 2)(x – 3) = 0，因此 x – 2 = 0 或 x – 3 = 0，所以 x = 2 或 x = 3。
Always expand your brackets mentally to confirm the middle term is correct; a very common error is a wrong sign in the factors. Also remember that after factorising, you must set each bracket equal to zero, not just write the factors.
务必在草稿上重新展开括号，确认中间项正确；因式分解中符号错误是非常常见的失分点。同时要记住，因式分解后必须令每个括号分别等于零，而不是只写出因式。
3. Completing the Square | 配方法
Completing the square is a powerful method because it works even when factorising is not obvious, and it reveals the turning point of the graph. For a monic quadratic x² + bx, the key identity is:
配方法是一种通用方法，即使因式分解不明显也能使用，而且能直接得到图像的顶点。对于首项系数为 1 的二次式 x² + bx，核心恒等式为：
x² + bx = (x + b/2)² – (b/2)²
Example: Express x² + 6x + 2 in completed-square form. Half of 6 is 3, and 3² = 9, so x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7.
例如，将 x² + 6x + 2 写成配方式。6 的一半是 3，3² = 9，因此 x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7。
If the coefficient of x² is not 1, first factor it out from the x² and x terms, complete the square inside the bracket, and then simplify. For instance, 2x² + 8x + 1 = 2(x² + 4x) + 1 = 2[(x + 2)² – 4] + 1 = 2(x + 2)² – 7.
若 x² 的系数不是 1，应先将该系数从 x² 项和 x 项中提出，在括号内完成配方，再化简。例如，2x² + 8x + 1 = 2(x² + 4x) + 1 = 2[(x + 2)² – 4] + 1 = 2(x + 2)² – 7。
4. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is obtained by completing the square on the general equation. The formula is:
求根公式可以解任意一元二次方程 ax² + bx + c = 0。它由一般形式配方得到。公式为：
x = [-b ± √(b² – 4ac)] / 2a
To use the formula, identify a, b and c carefully, substitute them with their signs, and simplify the square root. For example, 2x² + 3x – 2 = 0 gives x = [-3 ± √(3² – 4(2)(-2))] / (2(2)) = [-3 ± √25] / 4, so x = 1/2 or x = -2.
使用公式时，要先仔细确定 a、b、c，连同符号一起代入，再化简根号。例如，2x² + 3x – 2 = 0 中，x = [-3 ± √(3² – 4(2)(-2))] / (2(2)) = [-3 ± √25] / 4，所以 x = 1/2 或 x = -2。
When the answer is not exact, leave it in surd form unless the question asks for a decimal approximation. The ± sign is essential because there are usually two solutions.
当结果不是精确值时，除非题目要求小数近似，否则应保留根号形式。± 号必不可少，因为方程通常有两个解。
5. The Discriminant and Nature of Roots | 判别式与根的性质
The discriminant is the expression D = b² – 4ac under the square root in the quadratic formula. It tells us the nature of the roots without solving the equation.
判别式是求根公式中根号下的表达式 D = b² – 4ac。它可以在不解方程的情况下判断根的性质。
| Discriminant D | Nature of roots | Graph of y = ax² + bx + c |
| D > 0 | Two distinct real roots | Cuts the x-axis at two points |
| D = 0 | One repeated real root | Touches the x-axis at the turning point |
| D < 0 | No real roots | Does not cross the x-axis |
For example, x² – 6x + 9 = 0 has D = (-6)² – 4(1)(9) = 0, so it has one repeated root, x = 3. The equation x² + x + 2 = 0 has D = 1 – 8 = -7, so it has no real roots.
例如，x² – 6x + 9 = 0 的判别式 D = (-6)² – 4(1)(9) = 0，因此有一个重根 x = 3。方程 x² + x + 2 = 0 的 D = 1 – 8 = -7，因此没有实数根。
6. Quadratic Graphs: The Parabola | 二次函数图像：抛物线
The graph of y = ax² + bx + c is called a parabola. If a > 0, the parabola opens upwards and has a minimum turning point; if a < 0, it opens downwards and has a maximum turning point. The constant c is the y-intercept because y = c when x = 0.
二次函数 y = ax² + bx + c 的图像称为抛物线。若 a > 0，抛物线开口向上，有最低点；若 a < 0，开口向下，有最高点。常数 c 是 y 轴截距，因为当 x = 0 时 y = c。
To sketch a parabola, find the y-intercept, the x-intercepts if any, and the turning point. The shape is smooth and symmetric, not a V-shape or a broken line.
绘制抛物线时，应找出 y 轴截距、可能存在的 x 轴截距以及顶点。图像是光滑对称的曲线，不是 V 形或折线。
7. Turning Point and Axis of Symmetry | 顶点与对称轴
Completing the square gives y = a(x + p)² + q. The vertex is (-p, q), and the axis of symmetry is the vertical line x = -p. In standard form, the axis of symmetry can also be found using the formula:
配方法可把函数写成 y = a(x + p)² + q。顶点为 (-p, q)，对称轴为竖直直线 x = -p。在标准形式中，对称轴也可由以下公式求得：
x = -b / 2a
For example, y = x² – 4x + 3 can be written as y = (x – 2)² – 1, so the vertex is (2, -1) and the axis is x = 2. Since a = 1 > 0, this is a minimum point.
例如，y = x² – 4x + 3 可写成 y = (x – 2)² – 1，因此顶点为 (2, -1)，对称轴为 x = 2。因为 a = 1 > 0，该点为最小值点。
The turning point is useful for range problems, optimisation and sketching. Always state whether it is a maximum or minimum based on the sign of a.
顶点在求值域、优化问题和绘图时非常有用。作答时一定要根据 a 的符号说明它是最大值还是最小值。
8. Roots and x-Intercepts | 根与 x 轴截距
The real roots of ax² + bx + c = 0 are exactly the x-coordinates where the graph y = ax² + bx + c crosses the x-axis. If the graph does not cross the x-axis, the equation has no real roots.
方程 ax² + bx + c = 0 的实数根正是图像 y = ax² + bx + c 与 x 轴交点的横坐标。若图像不与 x 轴相交，则方程没有实数根。
This connection means you can solve an equation graphically by reading the x-intercepts, or check your algebraic solutions by plotting the graph. Repeated roots correspond to the graph touching the x-axis at one point only.
这种联系说明，可以通过读取 x 轴截距来图解方程，也可以借助图像检验代数解。重根对应图像在一点与 x 轴相切。
9. Using Graphs to Solve Equations | 利用图像解方程
A quadratic graph can be used to solve related equations. To solve ax² + bx + c = k, draw the horizontal line y = k and read the x-coordinates of intersection. To solve a quadratic equal to a linear expression, draw the straight line and find the points of intersection.
二次函数图像可用于解相关方程。要求解 ax² + bx + c = k，可画出水平直线 y = k，然后读出交点的横坐标。要求解二次式等于一次式，可画出该直线并找出交点。
Example: Given the graph of y = x² – 4x + 1, the equation x² – 4x + 1 = 0 is solved by reading the x-intercepts. The equation x² – 4x + 1 = 3 is solved by drawing y = 3 and reading the two intersection points.
例如，已知 y = x² – 4x + 1 的图像，方程 x² – 4x + 1 = 0 可通过读取 x 轴截距求解；方程 x² – 4x + 1 = 3 可通过画出 y = 3 并读取两个交点求解。
This method is especially common in IGCSE graph questions where you are given a printed grid and asked to solve modified equations without re-plotting the whole curve.
这种方法在 IGCSE 图像题中尤为常见：题目给出已画好的曲线和网格，要求解略作修改后的方程，而无需重新绘制整条曲线。
10. Applications and Word Problems | 应用题与文字题
Quadratic equations model many problems such as area, projectile motion, and number puzzles. Translate the problem into an equation, solve it, and reject solutions that do not make sense in context, such as negative lengths or negative time.
二次方程可以建立许多实际问题的模型，如面积、抛体运动和数字谜题。先把问题转化为方程，再求解，并舍去在情境中
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