📚 Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
A quadratic equation is one of the most important algebraic topics in IGCSE Mathematics. It appears in solving equations, drawing graphs, working with inequalities and modelling real-life problems. This article covers the standard form, factorising, the quadratic formula, completing the square, the discriminant, graphs and common exam mistakes.
二次方程是 IGCSE 数学中最重要的代数主题之一。它出现在解方程、画图像、处理不等式以及建立实际问题模型等题型中。本文涵盖标准形式、因式分解法、求根公式法、配方法、判别式、图像以及常见考试错误。
1. Recognising Quadratic Equations | 识别二次方程
If a = 0, the equation becomes linear rather than quadratic. The highest power of x is 2.
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。若 a = 0，方程就变成了一次方程。未知数 x 的最高次数为 2。
² − 5x + 6 = 0. Always collect all terms on one side and set the expression equal to zero.
在 IGCSE 题目中，你可能需要先把二次方程整理成标准形式再求解。例如 x² = 5x − 6 可化为 x² − 5x + 6 = 0。一定要把所有项移到一边，并让表达式等于 0。
2. Coefficients and Terminology | 系数与术语
icient, b is the coefficient of x, and c is the constant term. The sign of each term is part of the coefficient.
在 ax² + bx + c = 0 中，a 是二次项系数，b 是一次项系数，c 是常数项。每一项的正负号属于系数的一部分。
2 and c = 0. Misreading these signs is a common source of error in the quadratic formula.
对于 2x² − 3x + 4 = 0，a = 2，b = −3，c = 4。对于 −x² + 2x = 0，a = −1，b = 2，c = 0。在读系数时弄错正负号是使用求根公式时常见的错误来源。
3. Solving by Factorising | 因式分解法
Write the quadratic in the form (px + q)(rx + s) = 0, then set each bracket equal to zero.
当二次方程有较简单的根时，因式分解通常是最快的方法。把二次式写成 (px + q)(rx + s) = 0 的形式，然后令每个括号分别等于 0。
Example: Solve x² − 5x + 6 = 0. Factorise to (x − 2)(x − 3) = 0, so x = 2 or x = 3.
例如：解 x² − 5x + 6 = 0。因式分解为 (x − 2)(x − 3) = 0，因此 x = 2 或 x = 3。
s check by expanding the brackets to ensure the middle term and constant term are correct.
另一个例子：解 2x² + 5x − 3 = 0。因式分解为 (2x − 1)(x + 3) = 0，得到 x = 1/2 或 x = −3。一定要通过展开括号来检查中间项和常数项是否正确。
4. Solving by the Quadratic Formula | 求根公式法
The quadratic formula works for every quadratic equation, including those that do not factorise neatly:
求根公式适用于所有二次方程，包括不能整齐因式分解的方程：
x = (−b ± √b² − 4ac) ÷ (2a)
orm ax² + bx + c = 0. Substitute carefully with signs, especially when b or c is negative.
公式中的 a、b、c 来自标准形式 ax² + bx + c = 0。代入时要特别注意正负号，尤其是当 b 或 c 为负数时。
Example: Solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Then x = (4 ± √16 + 24) ÷ 4 = (4 ± √40) ÷ 4 = (2 ± √10) ÷ 2.
例如：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。则 x = (4 ± √16 + 24) ÷ 4 = (4 ± √40) ÷ 4 = (2 ± √10) ÷ 2。
When the question asks for exact answers, leave them in surd form. If a decimal approximation is required, use your calculator only at the final step.
当题目要求精确答案时，答案应保留根号形式。如果需要近似小数，应在最后一步再使用计算器。
5. Completing the Square | 配方法
Completing the square rewrites the quadratic in the form a(x + p)² + q. This form is useful for finding the turning point of a graph and for solving when factorising is difficult.
配方法把二次式写成 a(x + p)² + q 的形式。这种形式便于求图像顶点，也适用于难以因式分解的方程。
4. To solve x² + 6x + 5 = 0, we get (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.
例如：x² + 6x + 5 = (x + 3)² − 4。解 x² + 6x + 5 = 0 时，得到 (x + 3)² = 4，于是 x + 3 = ±2，解得 x = −1 或 x = −5。
square. For example, 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5.
当 a ≠ 1 时，要先对前两项提取 a，再完成配方。例如 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5。
6. Discriminant and Nature of Roots | 判别式与根的性质
he quadratic formula: Δ = b² − 4ac. It tells you how many real solutions the equation has.
判别式是求根公式中平方根内的表达式：Δ = b² − 4ac。它告诉我们方程有多少个实数解。
| Discriminant | 判别式 | Nature of roots | 根的性质 |
|---|---|
| Δ > 0 | Two distinct real roots | 两个不同实数根 |
| Δ = 0 | One repeated real root | 一个重根 |
| Δ < 0 | No real roots | 无实数根 |
xample: For x² − 4x + 4 = 0, Δ = (−4)² − 4(1)(4) = 0, so there is one repeated root x = 2.
例如：对于 x² − 4x + 4 = 0，Δ = (−4)² − 4(1)(4) = 0，因此有一个重根 x = 2。
, so the equation has no real roots. The graph of y = x² + x + 1 never crosses the x-axis.
例如：对于 x² + x + 1 = 0，Δ = 1² − 4(1)(1) = −3，小于 0，因此方程没有实数根。y = x² + x + 1 的图像不与 x 轴相交。
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。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。
, where the graph cuts or touches the x-axis. A quadratic with no real roots will not cross the x-axis at all.
y 轴截距是 c，因为当 x = 0 时 y = c。方程的根就是 x 轴截距，即图像与 x 轴相交或相切的横坐标。没有实数根的二次函数图像不会与 x 轴相交。
8. Turning Point and Axis of Symmetry | 顶点与对称轴
+ q gives the turning point at (−p, q). The axis of symmetry is the vertical line x = −p.
通过配方法，y = a(x + p)² + q 的顶点为 (−p, q)。对称轴是竖直直线 x = −p。
Example: y = (x − 3)² − 4 has a minimum at (3, −4) and axis of symmetry x = 3.
例如：y = (x − 3)² − 4 在 (3, −4) 处取得最小值，对称轴为 x = 3。
inate is 3, and the y-coordinate is (3)² − 6(3) + 5 = −4, so the turning point is (3, −4).
如果方程是标准形式，顶点的 x 坐标是 x = −b ÷ (2a)；将其代入函数即可求出 y 坐标。对于 y = x² − 6x + 5，x 坐标为 3，y 坐标为 (3)² − 6(3) + 5 = −4，因此顶点为 (3, −4)。
9. Solving Quadratic Inequalities | 二次不等式
To solve ax² + bx + c > 0 or ax² + bx + c < 0, first find the roots, then sketch the parabola. The sign of the quadratic changes at the roots.
解 ax² + bx + c > 0 或 ax² + bx + c < 0 时，先求出根，再画抛物线草图。二次式的符号在根处发生改变。
−2 and x = 3. Since a > 0, the graph is below the x-axis between the roots, so −2 < x < 3.
例如：解 x² − x − 6 < 0。根为 x = −2 和 x = 3。由于 a > 0，图像在两根之间位于 x 轴下方，因此 −2 < x < 3。
If the inequality were x² − x − 6 > 0, the solution would be x < −2 or x > 3. Always pay attention to whether the inequality is strict or includes equality.
如果不等式是 x² − x − 6 > 0，则解为 x < −2 或 x > 3。要注意不等式是否包含等号。
10. Word Problems and Applications | 应用题与实际应用
Quadratic equations often arise from area, projectile motion, profit and product problems. Define the unknown quantity clearly and form an equation from the given conditions.
二次方程常出现在面积、抛体运动、利润和乘积问题中。要清楚地设未知量，并根据条件建立方程。
1) = 42, giving n² + n − 42 = 0. Factorising gives (n + 7)(n − 6) = 0, so n = 6 or n = −7.
例如：两个连续整数的积为 42。设较小的整数为 n，则 n(n + 1) = 42，得 n² + n − 42 = 0。因式分解为 (n + 7)(n − 6) = 0，所以 n = 6 或 n = −7。
) = 40. Solve this as x² + 3x − 40 = 0 and then reject any negative solution for a length.
在几何题中，长为 (x + 3)、宽为 x 的长方形面积可能为 40，因此列出 x(x +
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