📚 Quadratic Equations: Methods and Graphs | 二次方程：解法与图像
ling. Any equation that can be written in the standard form ax² + bx + c = 0, where a ≠ 0, is a quadratic equation. Being able to solve it by factorising, completing the square or using the quadratic formula is an essential skill, and understanding its graph connects algebra to geometry.
二次方程贯穿 IGCSE 数学大纲，从代数运算到坐标几何和实际建模都有涉及。任何可以写成标准形式 ax² + bx + c = 0（其中 a ≠ 0）的方程都是二次方程。能够通过因式分解、配方法或求根公式解这类方程是一项基本技能，而理解它们的图像则把代数与几何联系起来。
1. What is a Quadratic Equation? | 什么是二次方程？
alled coefficients, and a is the coefficient of x², b is the coefficient of x, and c is the constant term.
一元二次方程是次数为 2 的方程：未知数的最高次数是 2。它的标准形式是 ax² + bx + c = 0，其中 a、b、c 是常数（称为系数），a 是 x² 的系数，b 是 x 的系数，c 是常数项。
Examples of quadratic equations include x² − 5x + 6 = 0, 2x² + 3x − 1 = 0 and x² − 9 = 0.
a ≠ 0 这一条件至关重要。如果 a 等于零，x² 项就会消失，方程就变成一次方程而不是二次方程。二次方程的例子包括 x² − 5x + 6 = 0、2x² + 3x − 1 = 0 和 x² − 9 = 0。
c once it is rearranged into standard form. Always expand brackets and collect all terms on one side before choosing a method.
有些二次方程看起来"不像"二次方程。例如 3x² = 12、(x + 2)(x − 5) = 0 或 x² = 4x − 3，只要整理成标准形式，它们仍然是二次方程。在选择题解法之前，应先把括号展开、把所有项移到同一边。
2. Roots, Solutions and the x-axis | 根、解与 x 轴
The solutions of a quadratic equation are called its roots. Since the highest power is 2, a quadratic can have two roots, one repeated root, or no real roots, depending on the values of the coefficients.
二次方程的解称为它的根。由于最高次数是 2，根据系数的取值，二次方程可能有两个根、一个重根，或者没有实数根。
o solving ax² + bx + c = 0 is exactly the same as asking where the curve meets the x-axis.
这些根对应着 y = ax² + bx + c 的图像与 x 轴交点的 x 值。x 轴上的点满足 y = 0，所以解 ax² + bx + c = 0 与求曲线与 x 轴交点完全等价。
The words "root", "solution" and "zero" are often used interchangeably in exams. When a question asks for the number of intersections with the x-axis, it is really asking for the number of real roots, which is decided by the discriminant.
在考试中，"根""解""零点"常常可以互换使用。当题目问与 x 轴的交点个数时，实际上就是在问实数根的个数，而这由判别式决定。
3. Solving by Factorising | 因式分解法
Factorising is the fastest method when the quadratic factorises into two brackets with integer or simple fractional terms. It relies on the zero product property: if the product of two factors is zero, then at least one of the factors must be zero.
当二次式能分解成两个含整数或简单分数的括号时，因式分解是最快的方法。它依据零乘积性质：如果两个因式的乘积为零，那么至少有一个因式等于零。
and add to −5, namely −2 and −3. Therefore x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
例如 x² − 5x + 6 = 0 可分解为 (x − 2)(x − 3) = 0。我们要找两个数，乘积为 +6、和为 −5，即 −2 和 −3。因此 x − 2 = 0 或 x − 3 = 0，得到 x = 2 或 x = 3。
3) = 2x² + 6x − x − 3 = 2x² + 5x − 3, so 2x − 1 = 0 or x + 3 = 0, giving x = ½ or x = −3.
当 x² 的系数不是 1 时，例如 2x² + 5x − 3 = 0，可以拆中间项或试算。这里 (2x − 1)(x + 3) = 2x² + 6x − x − 3 = 2x² + 5x − 3，所以 2x − 1 = 0 或 x + 3 = 0，得到 x = ½ 或 x = −3。
square, x² − 6x + 9 = (x − 3)². A perfect square gives a repeated root, here x = 3 twice.
有两个特殊结构值得记住：平方差 x² − 9 = (x − 3)(x + 3)，以及完全平方 x² − 6x + 9 = (x − 3)²。完全平方给出重根，这里 x = 3（二重根）。
4. Solving by Completing the Square | 配方法
Completing the square rewrites the quadratic as a perfect square plus or minus a constant. For a monic quadratic x² + bx, adding and subtracting (b ÷ 2)² creates a perfect square without changing the value of the expression.
配方法把二次式改写成一个完全平方加上或减去一个常数。对于首项系数为 1 的二次式 x² + bx，加上再减去 (b ÷ 2)² 就构造出完全平方，而不改变表达式的值。
x² + bx + c = (x + b ÷ 2)² − (b ÷ 2)² + c
x = 1. Notice the ± sign: both the positive and negative square roots must be considered.
例如 x² − 6x + 5 = 0 变为 (x − 3)² − 9 + 5 = 0，即 (x − 3)² − 4 = 0。于是 (x − 3)² = 4，所以 x − 3 = ±2，得到 x = 5 或 x = 1。注意 ± 号：正负两个平方根都必须考虑。
If a ≠ 1, first divide the whole equation by a, or factor a out of the first two terms. Completing the square is the method of choice when a question asks for the turning point, the maximum or minimum value, or an answer in exact surd form.
如果 a ≠ 1，先把整个方程除以 a，或从前两项中提出 a。当题目要求顶点坐标、最大值或最小值，或要求用根式精确表示答案时，配方法是最佳选择。
5. The Quadratic Formula | 求根公式
t factorises. It is derived by completing the square on the general form ax² + bx + c = 0.
求根公式可以解任何二次方程，无论它能否因式分解。它是对一般形式 ax² + bx + c = 0 使用配方法推导出来的。
x = (−b ± √b² − 4ac) ÷ (2a)
+ 3x − 1 = 0. Here a = 2, b = 3 and c = −1, so b² − 4ac = 9 + 8 = 17. Therefore x = (−3 ± √17) ÷ 4, which gives x ≈ 0.281 or x ≈ −1.781 to three decimal places.
例题：解 2x² + 3x − 1 = 0。这里 a = 2，b = 3，c = −1，所以 b² − 4ac = 9 + 8 = 17。因此 x = (−3 ± √17) ÷ 4，即 x ≈ 0.281 或 x ≈ −1.781（保留三位小数）。
Leave the answer as a surd if the question says "exact form", and only round at the very end if a decimal is requested. Substituting your answers back into the original equation is a quick and reliable check.
如果题目要求"精确形式"就把答案保留为根式；只有在要求小数时才在最后一步取近似值。把答案代回原方程检验，是快速而可靠的方法。
6. The Discriminant | 判别式
The expression under the square root, b² − 4ac, is called the discriminant and is often written as Δ. It tells you how many real roots the equation has before you do any solving.
根号下的表达式 b² − 4ac 称为判别式，常记作 Δ。在做任何求解之前，它就能告诉你方程有多少个实数根。
| Discriminant 判别式 | Nature of roots 根的性质 | Graph 图像 |
| Δ > 0 | Two distinct real roots 两个不同实根 | Crosses the x-axis twice 与 x 轴交于两点 |
| Δ = 0 | One repeated real root 一个重根 | Touches the x-axis 与 x 轴相切 |
| Δ < 0 | No real roots 无实数根 | Does not meet the x-axis 与 x 轴不相交 |
If Δ is a perfect square and a, b, c are integers, the roots are rational and the quadratic factorises over the integers. This is a useful link between the discriminant and the factorising method.
如果 Δ 是完全平方数，且 a、b、c 都是整数，那么根是有理数，二次式在整数范围内可以因式分解。这是判别式与因式分解法之间的一个有用联系。
oots, or a repeated root. Set Δ = k² − 36, then apply the relevant inequality or equality.
典型考题会问：k 取什么范围的值时，x² + kx + 9 = 0 有两个不同实根，或有重根。先写出 Δ = k² − 36，再根据相应条件列不等式或方程。
7. Graphs of Quadratic Functions: Parabolas | 二次函数图像：抛物线
. Its shape is determined by a: if a > 0 the parabola opens upwards and has a minimum point, while if a > 0 fails - that is, if a < 0 - the parabola opens downwards and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。它的形状由 a 决定：如果 a > 0，抛物线开口向上，有最小值点；如果 a < 0，抛物线开口向下，有最大值点。
m c gives the y-intercept, because substituting x = 0 gives y = c. The larger the value of |a|, the narrower the parabola; the smaller |a| is, the wider and flatter it looks.
常数项 c 给出 y 轴截距，因为令 x = 0 得 y = c。|a| 越大，抛物线越窄；|a| 越小，图像越宽、越平缓。
Every parabola is symmetric about a vertical line through its turning point. This symmetry means that the two roots, when they exist, are equally spaced on either side of the axis of symmetry.
每条抛物线都关于经过顶点的一条竖直线对称。这种对称性意味着：当两个根存在时，它们对称地分布在对称轴两侧，到对称轴的距离相等。
8. Vertex, Axis of Symmetry and Completed Square | 顶点、对称轴与配方式
turning point (vertex) lies on this line. Substituting that x-value back into the equation gives the y-coordinate of the vertex.
y = ax² + bx + c 的对称轴是竖直线 x = −b ÷ (2a)，顶点就在这条直线上。把该 x 值代回方程即可得到顶点的 y 坐标。
f symmetry is x = p. This is why completing the square is so powerful for graph questions.
当方程写成配方式 y = (x − p)² + q 时，顶点就是 (p, q)，对称轴是 x = p。这正是配方法在图像题中如此强大的原因。
−4, the y-intercept is 5, and the roots are x = 1 and x = 5 - both symmetric about x = 3.
例如 y = x² − 6x + 5 = (x − 3)² − 4。顶点是 (3, −4)，对称轴是 x = 3，最小值是 −4，y 轴截距是 5，根为 x = 1 和 x = 5，二者关于 x = 3 对称。
vertex: (1 + 5) ÷ 2 = 3. This shortcut is handy for checking your work quickly in an exam.
注意两根的平均值就是顶点的横坐标：(1 + 5) ÷ 2 = 3。这个技巧在考试中用于快速检查很有用。
9. Sketching and Interpreting Graphs | 画图与图像解读
A good sketch of a quadratic needs four features: the shape (up or down), the y-intercept, the x-intercepts (if any), and the coordinates of the turning point. Label each one clearly - examiners award marks for these labels.
一条合格的二次函数草图需要四个要素：开口方向、y 轴截距、x 轴截距（如果有）、顶点坐标。每一项都要清楚标注，阅卷时这些标注会给分。
hen Δ < 0. A curve that sits entirely above the x-axis with a > 0 has no real roots.
x 轴截距的个数与判别式一致：Δ > 0 时有两个交点，Δ = 0 时有一个，Δ < 0 时没有。当 a > 0 且整条曲线位于 x 轴上方时，方程没有实数根。
Graphs can also be used to solve inequalities. For example, x² − 6x + 5 < 0 is satisfied between the roots, so 1 < x < 5, while x² − 6x + 5 > 0 is satisfied outside the roots, so x < 1 or x > 5.
图像还可以用来解不等式。例如 x² − 6x + 5 < 0 在两根之间成立，即 1 < x < 5；而 x² − 6x + 5 > 0 在两根之外成立，即 x < 1 或 x > 5。
When a straight line and a parabola are drawn on the same axes, their intersection points are the solutions of the equation formed by setting the two expressions equal. This is a common "show that" style question.
当同一直角坐标系中同时画出直线与抛物线时，它们的交点就是令两个表达式相等所得方程的解。这是常见的"证明"型题目。
10. Modelling and Word Problems | 建模与应用题
Many real situations produce quadratic models: areas of rectangles with a fixed perimeter, projectile motion, braking distances and profit functions. The standard approach is to define a variable, form the equation, solve it, then interpret the answer in context.
许多实际情况会产生二次模型：周长固定时的矩形面积、抛体运动、刹车距离以及利润函数。标准做法是设未知数、列出方程、求解，然后结合情境解释答案。
r 4 seconds, and the maximum height occurs at t = 2, giving h = 20 × 2 − 5 × 2² = 40 − 20 = 20 metres.
例如：一个球被抛出后，t 秒时的高度（米）为 h = 20t − 5t²。令 h = 0 得 5t(4 − t) = 0，所以 t = 0 或 t = 4。球在空中停留 4 秒，最大高度出现在 t = 2 时，h = 20 × 2 − 5 × 2² = 40 − 20 = 20 米。
Always reject answers that make no physical sense, such as a negative length or a negative time. State clearly which root you are discarding and why - this earns a method mark.
一定要舍去不符合实际意义的答案，例如负长度或负时间。要明确说明舍弃哪个根以及原因，这样可以获得方法分。
s as (x + 8)(x − 5) = 0. Hence x = 5 (rejecting x = −8), so the rectangle is 5 cm by 8 cm.
例如：一个矩形的长比宽多 3 厘米，面积为 40 平方厘米。设宽为 x，则 x(x + 3) = 40，即 x² + 3x − 40 = 0，分解为 (x + 8)(x − 5) = 0。因此 x = 5（舍去 x = −8），矩形为 5 厘米 × 8 厘米。
11. Common Mistakes and Exam Tips | 常见错误与应试技巧
Most lost marks in quadratic questions come from a small number of repeated slips rather than from not knowing the method. Reading the question carefully and showing every step protects your marks.
二次方程题目的失分大多来自几个反复出现的低级错误，而不是不会方法。仔细读题并写出每一步，才能保住分数。
- Forgetting to rearrange into the form ax² + bx + c = 0 before factorising. — 因式分解前忘记整理成 ax² + bx + c = 0 的形式。
- Dividing both sides by x, which destroys the root x = 0. Always factorise instead. — 两边同除以 x，会丢掉 x = 0 这个根；应当使用因式分解。
- Losing the ± sign when taking a square root or using the formula. — 开平方或用公式时丢掉 ± 号。
- Sign errors when b or c is negative, especially inside b² − 4ac. — b 或 c 为负时符号出错，尤其是在 b² − 4ac 中。
- Rounding too early and carrying an inaccurate value through the working. — 过早取近似值，导致后续计算不准确。
- Giving decimal answers when the question asks for exact surd form. — 题目要求精确根式时却给出小数答案。
A reliable exam routine: identify a, b and c; check whether the expression factorises; if not, use the formula or complete the square; then verify by substitution. In graph questions, always label the intercepts and the turning point.
可靠的答题流程：先确定 a、b、c；判断表达式能否因式分解；若不能，则用公式或配方法；最后代入检验。图像题中，务必标注截距和顶点。
12. Quick Reference and Revision Checklist | 速查表与复习清单
Keep these key results in mind, because they cover almost every quadratic question that appears in an IGCSE paper.
牢记以下核心结论，它们几乎覆盖了 IGCSE 试卷中所有二次方程题目。
Standard form: ax² + bx + c = 0, a ≠ 0
Formula: x = (−b ± √b² − 4ac) ÷ (2a)
Discriminant: Δ = b² − 4ac
Axis of symmetry: x = −b ÷ (2a)
Completed square: x² + bx + c = (x + b ÷ 2)² − (b ÷ 2)² + c
Revision checklist: can you factorise a monic and a non-monic quadratic quickly? Can you complete the square and read off the vertex? Can you substitute into the formula without sign errors? Can you use the discriminant to count roots? Can you sketch a parabola and solve a quadratic inequality from it?
复习清单：你能否快速分解首项系数为 1 和不等于 1 的二次式？能否配方并直接读出顶点？能否代入公式而不出错符号？能否用判别式判断根的个数？能否画出抛物线并据此解二次不等式？
If you can answer yes
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