📚 Mastering Quadratic Equations | 掌握二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, graph sketching and real-life word problems, so a confident command of the solving methods and the discriminant is essential for top marks.
二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、坐标几何、图像绘制以及实际应用题中，因此熟练掌握各种解法与判别式是取得高分的关键。
This revision guide works through the standard form, the three main solving techniques, the discriminant, the shape of the parabola, quadratic inequalities, modelling questions and the errors that cost marks most often. Each section is written to match what examiners actually reward: a clear method, accurate algebra and a justified final answer.
本篇复习指南将依次讲解标准形式、三种主要解法、判别式、抛物线图像、二次不等式、建模应用题以及最常失分的错误。每一节都紧扣考官的评分要点：清晰的解题步骤、准确的代数运算以及有依据的最终答案。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be rearranged into the standard form below, where a, b and c are constants and a is not equal to zero. The highest power of the unknown is 2, which is what makes the equation "quadratic".
一元二次方程是可以整理成下面标准形式的方程，其中 a、b、c 是常数且 a 不等于零。未知数的最高次数是 2，这正是它被称为 "二次" 的原因。
ax² + bx + c = 0, a ≠ 0
- 2 = 0 and x² - 9 = 0 are all quadratic equations, because each one contains an x² term.
例如，x² - 5x + 6 = 0、2x² + 3x - 2 = 0 和 x² - 9 = 0 都是二次方程，因为每一个方程都含有 x² 项。
+ c = 0. Therefore the condition a ≠ 0 is essential for the equation to remain quadratic.
如果 a 等于零，x² 项就会消失，方程退化为一次方程 bx + c = 0。因此，a ≠ 0 这个条件是方程保持为二次方程的关键。
A quadratic equation has exactly two roots (solutions), although the two roots may be equal, and they may be irrational or not real at all - the discriminant in Section 6 tells you which case applies before you do any heavy algebra.
二次方程恰好有两个根（解），不过这两个根可能相等，也可能为无理数甚至不是实数——第 6 节的判别式能让你在做复杂运算之前就判断出属于哪种情况。
2. Standard Form and Identifying Coefficients | 标准形式与系数识别
Before using any method, you must rearrange the equation so that one side is exactly zero. Only then can you read off a, b and c correctly, and incorrect coefficients are the single most common source of lost marks.
在使用任何方法之前，必须先把方程整理成一边恰好为零的形式。只有这样才能正确读出 a、b、c 的值，而系数读错是最常见的失分原因。
-5 and c = -1. A missing term means the coefficient is zero, not that the term is absent.
例如，3x² = 5x + 1 必须先改写为 3x² - 5x - 1 = 0，才能得到 a = 3、b = -5、c = -1。某一项缺失意味着该项系数为零，而不是这一项不存在。
3x² - 5x - 1 = 0 → a = 3, b = -5, c = -1
The table below shows the coefficients for the examples used throughout this guide. Note how a sign travels with its term.
下表列出了本文全程使用的例子的系数。请注意符号是随着它所在的项一起移动的。
| Equation | 方程 | a | b | c |
|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | -5 | 6 |
| 2x² + 3x - 2 = 0 | 2 | 3 | -2 |
| x² - 9 = 0 | 1 | 0 | -9 |
When a is negative, multiply the whole equation by -1 first; it makes every later step easier and reduces sign errors dramatically.
当 a 为负数时，先对方程两边同乘 -1；这会让后续每一步都更简单，并大幅减少符号错误。
3. Solving by Factorising | 因式分解法
Factorising is the fastest method when it works. The logic depends on the zero product rule: if the product of two numbers is zero, at least one of them must be zero.
因式分解法在适用时是最快的方法。它的依据是零乘积法则：如果两个数的乘积为零，那么其中至少有一个数为零。
If (x - p)(x - q) = 0, then x = p or x = q
m is b. For x² - 5x + 6 = 0, the numbers -2 and -3 multiply to give +6 and add to give -5.
对于 a = 1 的二次方程，要寻找两个数，使它们的乘积等于 c、和等于 b。对于 x² - 5x + 6 = 0，数 -2 和 -3 相乘得 +6，相加得 -5。
x² - 5x + 6 = (x - 2)(x - 3) = 0 → x = 2 or x = 3
r 2x² + 3x - 2 = 0 the factorised form is (2x - 1)(x + 2), giving 2x - 1 = 0 or x + 2 = 0.
当 a 不等于 1 时，可以拆中间项，或者有系统地试算。对于 2x² + 3x - 2 = 0，分解结果为 (2x - 1)(x + 2)，于是有 2x - 1 = 0 或 x + 2 = 0。
2x² + 3x - 2 = (2x - 1)(x + 2) = 0 → x = ½ or x = -2
Special cases worth memorising are the difference of two squares and perfect squares:
值得记牢的特殊情况是平方差与完全平方：
- x² - 9 = (x - 3)(x + 3) = 0, so x = 3 or x = -3. | x² - 9 = (x - 3)(x + 3) = 0，所以 x = 3 或 x = -3。
- x² + 6x + 9 = (x + 3)² = 0, so x = -3 (a repeated root). | x² + 6x + 9 = (x + 3)² = 0，所以 x = -3（重根）。
- x² - 10x + 25 = (x - 5)² = 0, so x = 5 (a repeated root). | x² - 10x + 25 = (x - 5)² = 0，所以 x = 5（重根）。
4. Solving by Completing the Square | 配方法
Completing the square rewrites the quadratic as a perfect square plus a constant. It always works, it gives the vertex of the graph directly, and it is the only method examiners accept when a question says "by completing the square" or asks for the turning point.
配方法把二次式改写成 "完全平方加一个常数" 的形式。它总是可行，能直接给出图像的顶点，而且当题目要求 "用配方法" 或求顶点坐标时，它是考官唯一认可的方法。
x² + bx + c = (x + b/2)² - (b/2)² + c
efficient of x to get -3, square it to get 9, then subtract it again to keep the expression balanced.
以 x² - 6x + 5 = 0 为例。把 x 的系数除以 2 得到 -3，平方得 9，再把它减掉以保持表达式恒等。
(x - 3)² - 9 + 5 = 0 → (x - 3)² = 4
x - 3 = ±2 → x = 5 or x = 1
2x² + 8x + 3 = 0, write it as 2(x² + 4x) + 3, then complete the square inside the bracket.
当 a 不等于 1 时，先把 a 提到括号外。对于 2x² + 8x + 3 = 0，写成 2(x² + 4x) + 3，再在括号内配方。
2(x + 2)² - 8 + 3 = 2(x + 2)² - 5
k). Here the minimum point is (-2, -5), and because a = 2 > 0 the parabola opens upwards.
写成顶点式 y = a(x - h)² + k 后，顶点就是 (h, k)。此处最低点为 (-2, -5)，且因为 a = 2 > 0，抛物线开口向上。
5. Solving by the Quadratic Formula | 用求根公式解方程
The quadratic formula solves every quadratic equation, including those that do not factorise. Write it out in full before substituting, and substitute inside brackets to protect against sign errors.
求根公式可以解任何二次方程，包括那些不能因式分解的。代入之前先把公式完整写出，并且代入时加上括号以防符号出错。
x = ( -b ± √b² - 4ac ) ÷ (2a)
The expression underneath the square root, b² - 4ac, is called the discriminant. Evaluate it first, on its own line, then substitute.
根号下的表达式 b² - 4ac 称为判别式。先单独一行把它算出来，再进行代入。
Take 2x² + 3x - 2 = 0, so a = 2, b = 3 and c = -2:
以 2x² + 3x - 2 = 0 为例，a = 2、b = 3、c = -2：
b² - 4ac = 9 - 4 × 2 × (-2) = 9 + 16 = 25
x = ( -3 ± 5 ) ÷ 4 → x = ½ or x = -2
Now try x² - 4x + 1 = 0, which does not factorise with integers:
再看 x² - 4x + 1 = 0，它无法用整数因式分解：
x = ( 4 ± √12 ) ÷ 2 = ( 4 ± 2√3 ) ÷ 2 = 2 ± √3
Keep surds in exact form unless the question asks for a given number of decimal places. Rounding too early loses accuracy marks.
除非题目要求保留若干位小数，否则答案要保持精确的根式形式。过早取近似值会丢掉精度分。
6. The Discriminant | 判别式
The discriminant tells you how many real roots the equation has before you solve it, and it is a favourite in exam questions that say "find the value of k for which the equation has two equal roots".
判别式能让你在求解之前就判断方程有几个实根，也是考试中的常见考点，例如 "求使方程有两个相等实根的 k 值"。
Δ = b² - 4ac
| Value | 取值 | Nature of roots | 根的性质 | Graph | 图像 |
|---|---|---|
| Δ > 0 | Two distinct real roots | 两个不同的实根 | Crosses the x-axis twice | 与 x 轴交于两点 |
| Δ = 0 | One repeated real root | 一个重实根 | Touches the x-axis at the vertex | 与 x 轴相切于顶点 |
| Δ < 0 | No real roots | 没有实根 | Does not meet the x-axis | 与 x 轴不相交 |
, as expected. For x² + 6x + 9 = 0, Δ = 36 - 36 = 0, confirming a repeated root at x = -3.
对于 x² - 5x + 6 = 0，Δ = 25 - 24 = 1 > 0，所以两个根 2 和 3 不同，与预期一致。对于 x² + 6x + 9 = 0，Δ = 36 - 36 = 0，证实 x = -3 是重根。
here are no real solutions; the curve stays above the x-axis. A negative discriminant is a complete answer, not a failure.
对于 x² + x + 4 = 0，Δ = 1 - 16 = -15 < 0，因此没有实数解；曲线始终在 x 轴上方。判别式为负是一个完整的答案，不是做错了。
7. Roots, Intercepts and the Parabola | 根、截距与抛物线
. The sign of a controls the direction, the constant c gives the y-intercept, and the roots of the equation give the x-intercepts.
y = ax² + bx + c 的图像是一条抛物线。a 的符号决定开口方向，常数 c 给出 y 轴截距，而方程的根给出与 x 轴的交点。
- If a > 0, the parabola opens upwards and has a minimum point. | 若 a > 0，抛物线开口向上，有最低点。
- If a < 0, the parabola opens downwards and has a maximum point. | 若 a < 0，抛物线开口向下，有最高点。
- The y-intercept is at (0, c). | y 轴截距在 (0, c)。
- The axis of symmetry and the vertex both lie at x = -b ÷ (2a). | 对称轴和顶点都位于 x = -b ÷ (2a)。
x-vertex = -b ÷ (2a)
(0, 6), and the vertex is at x = 2.5, giving y = -0.25. The minimum point is (2.5, -0.25).
因此，画 y = x² - 5x + 6 的草图很快：根为 x = 2 与 x = 3，y 轴截距为 (0, 6)，顶点在 x = 2.5 处，对应 y = -0.25。最低点为 (2.5, -0.25)。
Examiners typically award one mark for each correct intercept, one for the correct shape and one for the labelled turning point.
考官通常给每个正确截距 1 分、正确形状 1 分，以及标注顶点 1 分。
8. Sum and Product of Roots | 两根之和与两根之积
you can find their sum and product without solving the equation at all. This is useful for checking answers and for questions about forming new equations.
若 ax² + bx + c = 0 的两根为 α 与 β，那么无需解方程就能求出它们之和与积。这对检查答案以及构造新方程的题目都很有用。
α + β = -b ÷ a, αβ = c ÷ a
the sum is -(-5) ÷ 1 = 5 and the product is 6 ÷ 1 = 6, and indeed 2 + 3 = 5 and 2 × 3 = 6.
用 x² - 5x + 6 = 0 验证：和为 -(-5) ÷ 1 = 5，积为 6 ÷ 1 = 6，而 2 + 3 = 5，2 × 3 = 6，完全吻合。
is -3 ÷ 2 = -1.5 and the product is -2 ÷ 2 = -1, matching ½ + (-2) = -1.5 and ½ × (-2) = -
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