📚 Quadratic Equations and Functions | 二次方程与函数
Quadratic equations and functions form one of the most important building blocks in the IGCSE Mathematics syllabus. You will encounter them in algebra, coordinate geometry, graph sketching, and a wide variety of real-world applications. Understanding the different methods of solution and the key features of the parabolic graph is essential for exam success.
二次方程与函数是IGCSE数学大纲中最重要的基础内容之一。从代数运算、坐标几何，到函数图像绘制和各类实际应用，你都会遇到它们。掌握不同的求解方法以及抛物线图像的关键特征，是考试取得高分的必备能力。
1. Standard Form of a Quadratic | 二次方程的标准形式
A quadratic equation in one variable can always be written in the standard form:
a x² + b x + c = 0, where a ≠ 0
equation becomes linear, so the condition a ≠ 0 defines a genuinely quadratic expression.
一元二次方程总可以写成标准形式：
a x² + b x + c = 0，其中 a ≠ 0
其中a是二次项系数，b是一次项系数，c是常数项。例如，3x² − 2x + 7 = 0 就是一个二次方程，其中 a = 3，b = −2，c = 7。如果a = 0，方程就退化为一次方程，因此条件a ≠ 0保证了表达式的二次性。
2. Solving by Factorisation | 因式分解法
When a quadratic expression can be factorised, this is often the fastest method. The principle is that if the product of two factors is zero, then at least one of the factors must be zero.
当一个二次表达式可以因式分解时，这通常是最快捷的解法。其原理是：若两个因式的乘积为零，则至少有一个因式为零。
look for two numbers that multiply to give 6 and add to give 5; these are 2 and 3. Hence:
例如，解方程 x² + 5x + 6 = 0。我们需要找到两个数，它们的乘积为6且和为5；这两个数就是2和3。因此：
(x + 2)(x + 3) = 0 → x = −2 or x = −3
(x + 2)(x + 3) = 0 → x = −2 或 x = −3
er. For 2x² − 5x − 3 = 0, we factor as (2x + 1)(x − 3) = 0, which gives x = −1/2 or x = 3.
解出答案后，务必在心里展开因式以验证结果。例如，解 2x² − 5x − 3 = 0 时，因式分解为 (2x + 1)(x − 3) = 0，因此 x = −1/2 或 x = 3。
3. The Quadratic Formula | 二次公式
dratic formula provides a universal method. For any quadratic equation a x² + b x + c = 0:
当因式分解困难或无法进行时，二次公式提供了一种通用解法。对于任意二次方程 a x² + b x + c = 0：
x = (−b ± √b² − 4ac) / (2a)
For example, solve 2x² + 3x − 5 = 0. Here a = 2, b = 3, c = −5. Substituting:
例如，解方程 2x² + 3x − 5 = 0。这里 a = 2，b = 3，c = −5。代入公式：
x = (−3 ± √9 + 40) / 4 = (−3 ± 7) / 4
ic formula always works, provided you substitute the values of a, b, and c carefully, especially when they are negative.
因此 x = 1 或 x = −5/2。二次公式始终有效，关键在于代入a、b、c的值时要仔细，尤其是当系数为负数时。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form p(x + q)² + r. This form reveals the turning point immediately and is also useful for solving equations.
配方法将二次式改写为 p(x + q)² + r 的形式。这种形式能直接显示图像的最高点或最低点，也常用于解方程。
For x² + 6x + 5 = 0, take half of 6, which is 3, and square it to get 9. Then:
以 x² + 6x + 5 = 0 为例，取6的一半得3，平方得9。于是：
x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4
g x = −1 or x = −5. In general, for x² + px + q = 0, we write (x + p/2)² − (p/2)² + q = 0.
令 (x + 3)² = 4，则 x + 3 = ±2，得到 x = −1 或 x = −5。一般地，对于 x² + px + q = 0，可写成 (x + p/2)² − (p/2)² + q = 0。
5. Graphs of Quadratic Functions | 二次函数的图像
x² + b x + c is a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
二次函数 y = a x² + b x + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，存在最低点；当 a < 0 时，抛物线开口向下，存在最高点。
rcepts at (0, −3) and roots at x = 3 and x = −1, so it crosses the x-axis at those points.
抛物线与y轴的交点恒为 (0, c)。与x轴的交点（若存在）就是方程 a x² + b x + c = 0 的实数根。例如，y = x² − 2x − 3 与y轴交于 (0, −3)，其根为 x = 3 和 x = −1，因此抛物线在这两点穿过x轴。
6. Vertex and Axis of Symmetry | 顶点与对称轴
(x − h)² + k, the vertex is at (h, k) and the axis of symmetry is the vertical line x = h.
抛物线的顶点可以通过配方法得到。对于 y = a(x − h)² + k，顶点坐标为 (h, k)，对称轴为竖直直线 x = h。
xis of symmetry x = 3. In standard form, the x-coordinate of the vertex is always −b/(2a).
例如，y = 2(x − 3)² + 5 的顶点为 (3, 5)，对称轴为 x = 3。在标准形式中，顶点的x坐标始终为 −b/(2a)。
xvertex = −b / (2a)
bstituting gives y = 1, so the vertex is (2, 1). Since a = 1 > 0, this point is a minimum.
要得到顶点的y坐标，只需将该x值代回函数。对于 y = x² − 4x + 5，顶点x坐标为2，代回得 y = 1，故顶点为 (2, 1)。由于 a = 1 > 0，该点是最低点。
7. The Discriminant | 判别式
dratic formula is called the discriminant. It tells us how many real roots the equation has without solving it completely.
二次公式中的 Δ = b² − 4ac 称为判别式。它可以在不完整求解的情况下判断方程有多少个实数根。
If Δ > 0: two distinct real roots, so the parabola crosses the x-axis twice.
若 Δ > 0：有两个不同的实数根，抛物线与x轴有两个交点。
0: one repeated real root, so the parabola touches the x-axis exactly once at its vertex.
若 Δ = 0：有一个重根，抛物线在顶点处与x轴相切。
If Δ < 0: no real roots, so the parabola never meets the x-axis.
若 Δ < 0：没有实数根，抛物线与x轴不相交。
one root, x = 2. The equation 2x² + x + 1 = 0 has Δ = 1 − 8 = −7, so it has no real roots.
例如，x² − 4x + 4 的判别式为 Δ = 16 − 16 = 0，因此它只有一个根 x = 2。方程 2x² + x + 1 = 0 的判别式为 Δ = 1 − 8 = −7，因此它没有实数根。
8. Sum and Product of Roots | 根的和与积
α and β are the roots of a x² + b x + c = 0, then there are two very useful relationships:
如果 α 和 β 是方程 a x² + b x + c = 0 的两个根，则有以下两个非常重要的关系：
α + β = −b / a (sum of roots)
α + β = −b / a（两根之和）
α β = c / a (product of roots)
α β = c / a（两根之积）
hese relationships allow quick checks and help in constructing equations from given roots.
例如，方程 x² − 5x + 6 = 0 的两个根为2和3，它们的和为5 = −(−5)/1，积为6 = 6/1。这些关系可用于快速检验答案，也可以在已知根的情况下构造对应的方程。
9. Solving Quadratic Equations Graphically | 图像法求近似解
nd read off the x-coordinates of the points where the graph crosses the x-axis. These x-intercepts are the solutions.
要用图像法解二次方程，可以绘制 y = a x² + b x + c 的图像，然后读出图像与x轴交点的x坐标。这些x截距就是方程的解。
those values. If the graph does not touch the x-axis, the equation has no real solutions.
例如，要用图像法解 x² − 2x − 3 = 0，先画出抛物线，观察它与x轴交于 x = −1 和 x = 3，即可写出解。如果图像不与x轴相交，则方程没有实数解。
Note that graphical solutions are usually approximate unless the roots happen to be integers. Always use an appropriate scale and plot enough points to draw a smooth curve.
注意，图像法得到的解通常是近似值，除非根恰为整数。绘图时须选择合适的比例尺，并描出足够多的点以便画出平滑的曲线。
10. Real-World Applications | 实际应用问题
relationship with time: h(t) = −g t² + v₀t + h₀. The maximum height occurs at the vertex.
二次函数可以建模许多现实情境。抛射体在重力作用下的高度与时间满足二次关系：h(t) = −g t² + v₀t + h₀。最大高度出现在顶点处。
econds. Substituting gives h(2) = −20 + 40 + 2 = 22 metres, so the maximum height is 22 m.
例如，h(t) = −5t² + 20t + 2 表示一个竖直上抛的小球。到达顶点的时间为 t = −20 / (2 × (−5)) = 2 秒。代回得 h(2) = −20 + 40 + 2 = 22 米，所以最大高度为22米。
Quadratics also appear in area problems, profit maximisation, and engineering design. In any such problem, the vertex represents either a maximum or minimum value, and roots often represent the boundaries of a feasible interval.
二次函数还出现在面积问题、利润最大化和工程设计中。在这类问题中，顶点代表最大值或最小值，而根通常代表可行区间的边界。
11. Common Pitfalls and Exam Tips | 常见误区与考试技巧
A common error is solving 3x² = 6x by dividing both sides by x and losing the root x = 0.
很多学生在二次方程上失分是因为本可避免的错误。首先，在应用任何解法之前，务必先将方程整理成标准形式 a x² + b x + c = 0。一个常见错误是解 3x² = 6x 时两边同除以x，从而丢失 x = 0 这个根。
Always rearrange to standard form first. | 总是先化为标准形式。
Never divide by a variable that could equal zero. | 切勿除以可能等于零的变量。
Check the sign of the discriminant to predict the number of roots. | 检查判别式的符号以预判根的个数。
Substitute your solutions back into the original equation to verify them. | 将解代回原方程进行验证。
When using the quadratic formula, be extra careful with negative values of b and c. | 使用二次公式时，要特别注意b和c为负数的情况。
For graph questions, always label the vertex, the y-intercept, and any x-intercepts clearly. | 对于图像题，务必清楚标注顶点、y截距和所有x截距。
Finally, practise drawing parabolas by hand using a small table of values, and memorise the completed-square form because it gives you the vertex instantly.
最后，多加练习用手工列表描点的方法绘制抛物线，并熟记配方法的形式，因为它能让你一目了然地看出顶点坐标。
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