📚 Quadratic Equations: Solving and Sketching | 二次方程：求解与图像
Quadratic equations are a major building block of IGCSE and A Level Mathematics. They appear in algebra, graphs, and real-life problem solving, and examiners often combine the algebraic method with a sketch or interpretation question, so you need to be fluent in both directions: from equation to graph, and from graph back to equation.
二次方程是 IGCSE 与 A Level 数学的重要基础内容。它们出现在代数、图像和实际应用问题中，考官经常将代数求解与图像草图或图形解释结合起来考查，因此你必须做到双向熟练：既能把方程变成图像，也能从图像反推出方程。
This article covers the definition of a quadratic, the three main solving methods (factorising, completing the square and the formula), the discriminant, and the standard technique for sketching a parabola accurately enough to earn full marks.
本文将系统讲解二次方程的定义、三大求解方法（因式分解法、配方法、求根公式）、判别式，以及能够拿到满分的抛物线作图标准技巧。
1. What Is a Quadratic Equation? | 什么是二次方程
A quadratic equation is an equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of x is 2, so the graph of the related function is a parabola.
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。x 的最高次数是 2，因此它所对应的函数图像是一条抛物线。
= 0 the term ax² disappears and you are left with bx + c = 0, which has only one solution.
系数 a 不能为零，因为那会使方程变成一次方程而不再是二次方程。如果 a = 0，ax² 这一项就消失了，剩下的 bx + c = 0 只有一个解。
It is important to separate two ideas that look almost identical. The equation ax² + bx + c = 0 has at most two solutions, called the roots. The function + c has a graph, and the roots of the equation are exactly the x-intercepts of that graph.
有两个看起来几乎一样、但必须区分清楚的概念：方程 ax² + bx + c = 0 最多有两个解，称为根；而函数 y = ax² + bx + c 有一条图像，方程的根恰好就是这条图像与 x 轴交点的横坐标。
ph, why the equation x² − 4x = k has two solutions when k is greater than a certain value.
这一联系正是考官偏爱二次方程的原因。题目可能先要求你解 x² − 4x = 0，然后要求你利用图像解释：为什么当 k 大于某个值时，方程 x² − 4x = k 有两个解。
2. Standard Form and Identifying a, b, c | 标准形式与识别 a、b、c
Before you factorise, complete the square or substitute into the formula, the equation must be in standard form with zero on the right-hand side. Rewriting is often where marks are lost, not the algebra itself.
在使用因式分解、配方或代入公式之前，方程必须整理成右边为零的标准形式。丢分往往发生在这个整理步骤，而不是代数运算本身。
2. Subtract 5x and 2 from both sides to get 3x² − 5x − 2 = 0, so a = 3, b = −5 and c = −2.
例题：解 3x² = 5x + 2。两边同时减去 5x 和 2，得到 3x² − 5x − 2 = 0，所以 a = 3，b = −5，c = −2。
gn errors here destroy the rest of the calculation, so write down a, b and c explicitly every single time.
要特别注意 b = −5 而不是 5。公式中用的是 −b，而 −(−5) = +5。这里的符号错误会毁掉后面的全部计算，所以每一次都要把 a、b、c 明确写出来。
0 if you multiply throughout by −1. Multiplying by −1 is usually the safer choice because a positive a makes completing the square and sketching easier.
还要留意没有按降幂排列的二次方程，例如 7 − x² = 4x。整理后得到 −x² − 4x + 7 = 0；如果两边同乘以 −1，则得到等价的 x² + 4x − 7 = 0。通常乘 −1 更稳妥，因为 a 为正时配方和作图都更方便。
3. Solving by Factorising | 因式分解法
When the quadratic can be factorised, write it as (px + q)(rx + s) = 0. Then set each bracket equal to zero and solve for x. This works because if the product of two numbers is zero, at least one of them must be zero.
当二次式可以因式分解时，把它写成 (px + q)(rx + s) = 0。然后令每个括号分别等于零并解出 x。这条方法的依据是：如果两个数的乘积为零，那么其中至少有一个数为零。
Example: x² − 5x + 6 = 0 factorises to (x − 2)(x − 3) = 0, so x = 2 or x = 3.
示例：x² − 5x + 6 = 0 因式分解为 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。
Always check by expanding the brackets. If the middle term has the wrong sign, check the signs inside the brackets. For x² − 5x + 6, the constant term +6 is positive so both numbers have the same sign, and the middle term −5x is negative so both must be negative: −2 and −3.
一定要展开括号进行检验。如果中间项的符号不对，就检查两个括号里的符号。对于 x² − 5x + 6，常数项 +6 为正，说明两个数同号；中间项 −5x 为负，说明两个数都为负：即 −2 和 −3。
ethod is: find two numbers whose product is c and whose sum is b. A quick table keeps this organised:
对于 a = 1 的简单二次式，方法是：找出两个数，使它们的乘积等于 c、和等于 b。用一个小表格可以理清思路：
| Equation | Product = c | Sum = b | Factorised |
| x² + 7x + 12 = 0 | +12 | +7 | (x + 3)(x + 4) = 0 |
| x² − x − 20 = 0 | −20 | −1 | (x − 5)(x + 4) = 0 |
| x² − 9x + 20 = 0 | +20 | −9 | (x − 4)(x − 5) = 0 |
hey are 1 and 6), split the middle term, then factorise in pairs to obtain (2x + 1)(x + 3) = 0, giving x = −½ or x = −3.
当 a ≠ 1 时，例如 2x² + 7x + 3 = 0，需要用尝试法或 ac 法：先算 a × c = 6，找出乘积为 6、和为 7 的两个数（1 和 6），把中间项拆开，再分组分解，得到 (2x + 1)(x + 3) = 0，所以 x = −½ 或 x = −3。
0 has no integer factors, so be ready to switch to completing the square or the quadratic formula.
请记住，并非所有二次式都能在整数范围内因式分解。x² + x − 1 = 0 就没有整数因子，因此要随时准备改用配方法或求根公式。
4. Completing the Square | 配方法
Completing the square turns any quadratic into the form a(x − h)² + k. It is the most reliable method when factorising fails, and it is the only method that gives you the vertex of the parabola directly.
配方能把任意二次式化成 a(x − h)² + k 的形式。它是在因式分解失效时最可靠的方法，也是唯一能直接给出抛物线顶点的方法。
The key identity is:
核心恒等式是：
x² + bx = (x + b ÷ 2)² − (b ÷ 2)²
e x² + 6x = (x + 3)² − 9. The equation becomes (x + 3)² − 9 + 1 = 0, that is (x + 3)² = 8.
例题：解 x² + 6x + 1 = 0。6 的一半是 3，所以把 x² + 6x 写成 (x + 3)² − 9。方程变为 (x + 3)² − 9 + 1 = 0，即 (x + 3)² = 8。
Taking square roots of both sides gives x + 3 = ±√8 = ±2√2, so x = −3 ± 2√2, which is approximately x = −0.172 or x = −5.828.
两边开平方得 x + 3 = ±√8 = ±2√2，所以 x = −3 ± 2√2，约为 x = −0.172 或 x = −5.828。
s two values, and losing one of them loses one of the two roots, and usually a mark as well.
千万不要丢掉 ± 号。平方根有两个值，丢掉其中一个就等于丢掉两个根中的一个，通常也会丢掉分数。
− 5. Notice that the constant term is outside the bracket and must not be multiplied by 2.
如果 a ≠ 1，只需从前两项中提出 a。例如 2x² − 8x + 3 = 2(x² − 4x) + 3 = 2[(x − 2)² − 4] + 3 = 2(x − 2)² − 5。注意常数项在括号外面，不要被 2 乘到。
5. The Quadratic Formula | 二次方程求根公式
The quadratic formula solves every quadratic equation, factorisable or not:
求根公式可以解出任何二次方程，无论它能否因式分解：
x = (−b ± √b² − 4ac) ÷ 2a
² + 5x − 3 = 0. Here a = 2, b = 5, c = −3, so b² − 4ac = 5² − 4 × 2 × (−3) = 25 + 24 = 49.
例题：解 2x² + 5x − 3 = 0。这里 a = 2，b = 5，c = −3，所以 b² − 4ac = 5² − 4 × 2 × (−3) = 25 + 24 = 49。
Then √49 −3. Expanding (2x − 1)(x + 3) confirms the answer, since 2x² + 5x − 3 = 0 does factorise.
于是 √49 = 7，x = (−5 ± 7) ÷ 4，得到 x = 0.5 或 x = −3。展开 (2x − 1)(x + 3) 可以验证结果，因为 2x² + 5x − 3 = 0 确实可以因式分解。
Three practical rules save marks with the formula. First, write down a, b and c before substituting. Second, calculate the discriminant b² − 4ac as a separate step, because many errors happen inside the square root. Third, keep full calculator accuracy until the final answer, then round to the accuracy the question asks for, usually 3 significant figures.
使用公式时有三条实用规则可以保分。第一，代入之前先写出 a、b、c。第二，把判别式 b² − 4ac 单独算一步，因为很多错误都发生在根号里面。第三，计算过程中保留计算器的完整精度，最后再按题目要求取近似值，通常是 3 位有效数字。
If the question says "give your answers to 3 significant figures", examiners expect the unsimplified surd form as well when the roots are irrational, unless the question explicitly says otherwise.
如果题目要求"答案保留 3 位有效数字"，当根是无理数时，考官通常希望你也写出未化简的根式形式，除非题目另有明确说明。
6. The Discriminant: How Many Roots? | 判别式：根的个数
The expression under the square root, b² − 4ac, is called the discriminant, often written as Δ. It tells you how many real roots the equation has without solving it.
根号下面的表达式 b² − 4ac 称为判别式，常记作 Δ。它可以在不解方程的情况下告诉你方程有多少个实根。
| Value of Δ | Number of real roots | Graph meets x-axis |
| Δ > 0 | two distinct real roots | twice |
| Δ = 0 | style="
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