📚 Mastering Quadratic Equations and Parabolic Graphs | 掌握二次方程与抛物线图像
Quadratic equations form one of the largest topics in IGCSE and A-Level Mathematics. They are tested through pure algebra, coordinate geometry, function analysis and real-life modelling, and they reappear later inside calculus, mechanics and statistics. This guide brings together the methods you need in order to solve any quadratic and to sketch its graph accurately.
二次方程是 IGCSE 与 A-Level 数学中最重要的主题之一，考查形式包括纯代数运算、坐标几何、函数分析和现实建模，并且会在后续的微积分、力学与统计中反复出现。本指南汇总了求解任意二次方程以及准确绘制其图像所需的方法。
1. Recognising a Quadratic Expression | 识别二次表达式
A quadratic expression has the standard form ax² + bx + c, where a, b and c are constants and a ≠ 0. The key test is that the highest power of x is exactly 2, not 3 or any higher power, and x cannot appear in a denominator or under a square root.
二次表达式的标准形式为 ax² + bx + c，其中 a、b 和 c 是常数，且 a ≠ 0。判断的关键在于 x 的最高次数必须正好是 2，不能是 3 或更高次数，x 也不能出现在分母中或平方根号下。
For example, 4x² − 3x + 7 is quadratic because the term with the highest power is 4x². The expression 2x³ + x² − 1 is not quadratic because it contains x³. The expression 1/x + x is also not quadratic because division by x breaks the polynomial condition.
例如，4x² − 3x + 7 是二次式，因为最高次项是 4x²。2x³ + x² − 1 不是二次式，因为它含有 x³。1/x + x 也不是二次式，因为除以 x 破坏了多项式条件。
6 − x), or as (x − 1)² + 5, or given through a table of values. Always expand or rearrange first, then decide whether the degree is 2.
在考题中，二次式常常被隐藏起来，可能写成 y = x(6 − x)，或者 (x − 1)² + 5，或者通过数值表给出。一定要先展开或整理，再判断次数是否为 2。
ax² + bx + c, a ≠ 0
2. Expanding and Factorising Brackets | 展开与因式分解
(p + q)x + pq is very useful because factorising is simply the reverse of this expansion.
展开两个二项式使用分配律。一般结果 (x + p)(x + q) = x² + (p + q)x + pq 非常有用，因为因式分解正是这一展开过程的逆运算。
(x + p)(x + q) = x² + (p + q)x + pq
s 2 and 5 work because 2 × 5 = 10 and 2 + 5 = 7, so the factorised form is (x + 2)(x + 5).
在对 x² + bx + c 进行因式分解时，需要找到两个数，它们的乘积为 c，和为 b。对于 x² + 7x + 10，数字 2 和 5 满足条件，因为 2 × 5 = 10，且 2 + 5 = 7，因此因式分解结果为 (x + 2)(x + 5)。
sum is 11 (namely 21 and −10), split the middle term, then factorise in pairs. Alternatively, use the formula x = (−b ± √b² − 4ac) / 2a to find the roots and work backwards.
当 x² 的系数不是 1 时，例如 6x² + 11x − 35，可使用 ac 法：先算 a 与 c 的乘积得 −210，再找两个数使其乘积为 −210、和为 11（即 21 与 −10），然后拆开中间项、分组因式分解。也可以先用求根公式 x = (−b ± √b² − 4ac) / 2a 求出根，再反推因式。
- Always check for a common factor first: 3x² − 12 = 3(x − 2)(x + 2). | 先提取公因式：3x² − 12 = 3(x − 2)(x + 2)。
- Difference of two squares: a² − b² = (a + b)(a − b). | 平方差公式：a² − b² = (a + b)(a − b)。
- Perfect square: a² ± 2ab + b² = (a ± b)². | 完全平方：a² ± 2ab + b² = (a ± b)²。
3. Solving by Factorisation | 因式分解法求解
e form ax² + bx + c = 0, factorise the left side, and then set each bracket equal to zero.
如果二次方程可以写成乘积等于零的形式，就可以快速求解。先将方程整理为 ax² + bx + c = 0，再对左边进行因式分解，然后令每个括号分别等于零。
5x + 6 = 0 becomes (x + 2)(x + 3) = 0, so x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.
例如，x² + 5x + 6 = 0 可写成 (x + 2)(x + 3) = 0，于是 x + 2 = 0 或 x + 3 = 0，解得 x = −2 或 x = −3。
This relies on the zero product property: if the product of two real numbers is zero, at least one of them must be zero. Never divide both sides by x, because x may equal zero and you would lose a root.
这一方法依赖零乘积性质：若两个实数的乘积为零，则其中至少有一个为零。千万不要在两边同除以 x，因为 x 可能等于零，那样会丢失一个根。
- Step 1: rearrange so that the right side is exactly zero. | 第一步：整理方程，使右边恰好为零。
- Step 2: take out common factors and factorise fully. | 第二步：提取公因式并彻底分解。
- Step 3: set each factor to zero and solve the linear equations. | 第三步：令每个因式为零，解出这两个一元一次方程。
- Step 4: substitute back to check both answers. | 第四步：代回原方程检验两个答案。
4. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This form reveals the vertex of the parabola immediately and is the standard method for proving the quadratic formula.
配方法把 ax² + bx + c 改写为 a(x + p)² + q 的形式。这种形式能立刻显示抛物线的顶点，也是证明求根公式的标准方法。
x² + bx + c = (x + b/2)² − (b/2)² + c
3, so write (x + 3)² − 9 + 2 = 0, that is (x + 3)² = 7. Taking square roots gives x = −3 ± √7, so x ≈ −0.354 or x ≈ −5.646.
例题：x² + 6x + 2 = 0。6 的一半是 3，所以写成 (x + 3)² − 9 + 2 = 0，即 (x + 3)² = 7。开平方得 x = −3 ± √7，所以 x ≈ −0.354 或 x ≈ −5.646。
rst two terms only: 2x² + 12x + 5 = 2(x² + 6x) + 5 = 2[(x + 3)² − 9] + 5 = 2(x + 3)² − 13.
当 a ≠ 1 时，只需从前两项中提取 a：2x² + 12x + 5 = 2(x² + 6x) + 5 = 2[(x + 3)² − 9] + 5 = 2(x + 3)² − 13。
5. The Quadratic Formula and the Discriminant | 求根公式与判别式
When factorisation is not possible with integers, the quadratic formula always works. It is worth memorising exactly, including the division by 2a rather than a.
当因式分解无法用整数完成时，求根公式总是有效。它值得精确记忆，包括分母是 2a 而不是 a。
x = (−b ± √b² − 4ac) / 2a
The expression b² − 4ac is called the discriminant and is written Δ. Its sign tells you how many real roots the equation has, which in turn tells you how many times the parabola crosses the x-axis.
表达式 b² − 4ac 称为判别式，记作 Δ。它的正负决定了方程有多少个实根，也决定了抛物线与 x 轴相交多少次。
| Δ = b² − 4ac | Number of real roots | 实根个数 | Graph behaviour | 图像表现 |
| Δ > 0 | two distinct roots | 两个不同实根 | crosses the x-axis twice | 与 x 轴交于两点 |
| Δ = 0 | one repeated root | 一个重根 | touches the x-axis at the vertex | 顶点与 x 轴相切 |
| Δ < 0 | no real roots | 无实根 | never reaches the x-axis | 完全不接触 x 轴 |
3. Always simplify the surd and, if asked for exact answers, leave them in surd form rather than decimal form.
例如：3x² − 5x − 2 = 0 得 Δ = 25 + 24 = 49，于是 x = (5 ± 7) / 6，即 x = 2 或 x = −1/3。一定要把根式化简；若题目要求精确值，应保留根式形式而不是写成小数。
6. The Shape of a Parabola | 抛物线的形状
. It is a smooth symmetric curve with a single turning point, and every vertical line meets it at most twice.
y = ax² + bx + c 的图像是一条抛物线。它是一条光滑的对称曲线，只有一个转向点，并且任意一条竖直线与它最多相交两次。
The sign of a controls the direction. If a > 0 the parabola opens upwards and the vertex is a minimum point. If a < 0 the parabola opens downwards and the vertex is a maximum point.
a 的符号决定开口方向。若 a > 0，抛物线开口向上，顶点为最低点；若 a < 0，抛物线开口向下，顶点为最高点。
ide, flat curve. Comparing y = 4x² and y = x², the first is four times steeper everywhere.
a 的绝对值决定宽窄。例如 a = 5 这类较大的值使曲线变得很窄，而 a = 0.2 这类较小的值使曲线又宽又平。比较 y = 4x² 与 y = x²，前者的每一处都比后者陡峭四倍。
7. Vertex, Axis of Symmetry and Intercepts | 顶点、对称轴与截距
The axis of symmetry is the vertical line x = −b / (2a), and the vertex lies on this line.
抛物线的每一个关键特征都能从方程中读出。对称轴是竖直线 x = −b / (2a)，而顶点就位于这条直线上。
Axis of symmetry: x = −b / (2a) Vertex: ( −b / (2a), c − b² / (4a) )
To find the y-coordinate of the vertex, substitute the x-value of the axis back into the original equation. This is usually safer than memorising the vertex formula.
要确定顶点的纵坐标，把对称轴的 x 值代回原方程即可。这通常比记忆顶点公式更保险。
- y-intercept: let x = 0, so y = c. The curve always crosses the y-axis at (0, c). | y 轴截距：令 x = 0，得 y = c，曲线必定经过点 (0, c)。
- quadratic. These are the roots found by factorising, completing the square or the formula. | x 轴截距：令 y = 0，解二次方程，其解就是前面几种方法求得的根。
- If Δ < 0 there is no x-intercept, and the whole curve lies above or below the x-axis. | 若 Δ < 0，则没有 x 轴截距，整条曲线都位于 x 轴上方或下方。
8. Sketching a Parabola from Its Equation | 由方程绘制抛物线草图
A sketch does not need to be perfectly to scale, but it must show shape, direction and labelled key points. Examiners award marks for these features, not for artistic quality.
草图不需要严格按比例绘制，但必须体现形状、开口方向以及标注清楚的关键点。评分标准给分的是这些要素，而不是画得好不好看。
of symmetry is the midpoint x = 2, and substituting gives y = 4 − 8 + 3 = −1, so the vertex is (2, −1). The y-intercept is (0, 3), which by symmetry mirrors to (4, 3).
绘图示例：y = x² − 4x + 3。这里 a = 1 > 0，抛物线开口向上。因式分解得 (x − 1)(x − 3) = 0，所以与 x 轴交于 1 和 3。对称轴在中点 x = 2，代入得 y = 4 − 8 + 3 = −1，故顶点为 (2, −1)。y 轴截距为 (0, 3)，由对称性可得对称点 (4, 3)。
- Step 1: decide the direction from the sign of a. | 第一步：由 a 的符号判断开口方向。
- Step 2: find the roots to locate the x-intercepts, if any. | 第二步：求根，确定 x 轴截距（若存在）。
- Step 3: find the vertex and draw the axis of symmetry as a dashed line. | 第三步：求顶点，用虚线画出对称轴。
- Step 4: mark the y-intercept and label all coordinates. | 第四步：标出 y 轴截距，并标注所有坐标。
9. Transformations of Parabolic Graphs | 抛物线图像的变换
basic curve y = x² has simply been translated h units horizontally and k units vertically.
配方法形式 y = a(x − h)² + k 让图像变换一目了然。顶点是 (h, k)，也就是说基本曲线 y = x² 被水平平移 h 个单位、竖直平移 k 个单位。
(x + 3)² moves it 3 units to the left. Outside the bracket, +k moves the curve up and −k moves it down.
要特别注意符号约定：括号内，y = (x − 3)² 把曲线向右平移 3 个单位，而 y = (x + 3)² 把曲线向左平移 3 个单位。括号外，+k 向上平移，−k 向下平移。
turning every minimum into a maximum. The equation y = −x² is the mirror image of y = x².
乘以负数会关于 x 轴作反射，把最低点变成最高点。方程 y = −x² 就是 y = x² 关于 x 轴的镜像。
10. Word Problems and Modelling | 应用题与建模
Many exam questions place quadratics inside a real context: the height of a thrown ball, the area of a rectangle with a fixed perimeter, or the profit of a small business. The mathematical steps stay the same, but the answer must make physical sense.
许多考题把二次方程放进真实情境：抛出小球的高度、固定周长下矩形的面积，或小生意的利润。数学步骤不变，但答案必须符合实际意义。
x². The maximum area occurs at the vertex, x = 5, giving A = 25 cm², which is the square.
典型题目：矩形周长为 20 cm。设宽为 x，则长为 10 − x，面积为 A = x(10 − x) = 10x − x²。最大面积出现在顶点处，x = 5，得 A = 25 cm²，此时恰好是正方形。
0 and, in most questions, only up to the positive root where h returns to ground level. Reject any negative time.
对于抛体运动，例如 h = −5t² + 20t + 1，模型只在 t ≥ 0 时有效，而且多数题目中只取 h 回到地面时的正根。负的时间必须舍去。
11. Common Exam Mistakes | 常见考试错误
Most lost marks in quadratic questions come from a small set of recurring errors. Checking these four points before you move on can save several marks in a single paper.
二次方程题目中丢分大多来自少数几个反复出现的错误。在继续下一题之前检查以下四点，可能在一份试卷中挽回好几分。
- Forgetting the ± sign in the formula, and so reporting only one root when there are two. | 漏掉公式中的 ± 号，明明有两个根却只写一个。
- Dividing by a instead of 2a in the quadratic formula. | 使用求根公式时除以 a 而不是 2a。
- Sign errors when b or c is negative, especially with −b and with (b/2)² in completing the square. | b 或 c 为负数时的符号错误，尤其是 −b 和配方中的 (b/2)²。
- g to ax² + bx + c = 0 before factorising, so the factors are set equal to the wrong value. | 未先整理成 ax² + bx + c = 0 就分解，导致因式被设为错误的值。
Another frequent issue is rounding too early. Keep full accuracy through the working and round only at the final line, normally to three significant figures unless the question states otherwise.
另一个常见问题是过早取近似值。中间过程要保持完整精度，只在最后一步取近似值，通常保留三位有效数字，除非题目另有说明。
12. Exam Technique and Practice Plan | 应试技巧与复习计划
Quadratic questions are usually marked in stages: method marks for the correct formula or factorisation, accuracy marks for correct roots, and interpretation marks for choosing the right answer in context. Show every step, even when you can do it mentally.
二次方程题通常分步给分：正确使用公式或因式分解得方法分，根算对得准确分，在情境中选择合理答案得解释分。即使你能心算，也要写出每一步。
Use the discriminant as a quick check. If you find two roots but Δ < 0, something is wrong. If your sketch crosses the x-axis twice but your roots are equal, something is wrong. This self-checking habit is what separates a grade A from a grade A*.
用判别式快速自查。若你求出两个根但 Δ < 0，那一定有错；若图像与 x 轴交于两点但你的两个根相等，也一定有错。这种自我检查的习惯正是 A 与 A* 之间的差别。
- Week 1: expanding, factorising and the ac method. | 第一周：展开、因式分解与 ac 法。
- Week 2: completing the square and the quadratic formula. | 第二周：配方法与求根公式。
- Week 3: parabola sketching, vertex and intercepts. | 第三周：抛物线草图、顶点与截距。
- Week 4: past-paper questions under timed conditions. | 第四周：限时完成历年真题。
Master these twelve sections and quadratic equations stop being a memory test and become a reliable source of marks in every paper you sit.
掌握这十二个部分之后，二次方程就不再是背诵测试，而会成为你每一份试卷中稳定可靠的得分来源。
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