📚 Mastering Quadratic Equations | 掌握二次方程
Quadratic equations appear in nearly every IGCSE Mathematics paper, whether in algebra, graphs, or problem-solving questions. This guide walks through the standard form, factorisation, completing the square, the quadratic formula, the discriminant, and graph sketching, so you can approach any quadratic question with confidence.
二次方程几乎出现在每一份 IGCSE 数学试卷中，无论是代数、图象还是应用题都会涉及。本指南系统讲解标准形式、因式分解、配方法、求根公式、判别式以及图象画法，帮助你有信心应对任何二次方程问题。
1. The Standard Form of a Quadratic | 二次方程的标准形式
e highest power of x is exactly 2. If a = 0, the expression becomes linear, not quadratic.
二次表达式的一般形式为 ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。x 的最高次数正好是 2。若 a = 0，表达式就变成一次式，而不是二次式。
Examples of quadratic expressions include 2x² + 3x − 5, x² − 4, and 3x² + 7x. Expressions such as 2x + 1 or x³ + x² are not quadratic.
二次表达式的例子包括 2x² + 3x − 5、x² − 4 和 3x² + 7x。而 2x + 1 或 x³ + x² 这样的式子则不是二次的。
The coefficient a controls the shape of the graph. When a > 0, the parabola opens upward like a smile; when a < 0, it opens downward like a frown.
系数 a 决定图象的形状。当 a > 0 时，抛物线开口向上，如同笑脸；当 a < 0 时，开口向下，如同倒过来的笑脸。
2. Expanding and Factorising | 展开与因式分解
rackets. The most common form is x² + bx + c = (x + p)(x + q), where p + q = b and pq = c.
因式分解就是把二次式写成两个一次括号相乘。最常见的形式是 x² + bx + c = (x + p)(x + q)，其中 p + q = b，pq = c。
ck your factorisation by expanding again: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.
例如，x² + 5x + 6 = (x + 2)(x + 3)，因为 2 + 3 = 5，且 2 × 3 = 6。务必通过再次展开来检验因式分解是否正确：(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6。
two squares is a² − b² = (a − b)(a + b), and a perfect square is a² + 2ab + b² = (a + b)².
有两个特殊结论值得牢记。平方差公式：a² − b² = (a − b)(a + b)；完全平方公式：a² + 2ab + b² = (a + b)²。
where m × n = a and p × q = c. Testing combinations gives (2x + 1)(x + 3) = 2x² + 7x + 3.
当 x² 的系数不是 1 时，例如 2x² + 7x + 3，需要寻找形如 (mx + p)(nx + q) 的两个括号，其中 m × n = a，p × q = c。尝试组合可得 (2x + 1)(x + 3) = 2x² + 7x + 3。
3. Solving by Factorisation | 因式分解法求解
se the left-hand side and apply the zero-product property: if AB = 0, then A = 0 or B = 0.
求解二次方程时，首先将方程整理为一侧为零：ax² + bx + c = 0。然后对左侧进行因式分解，并运用零乘积性质：若 AB = 0，则 A = 0 或 B = 0。
ng each bracket to zero gives x − 2 = 0 or x − 3 = 0, so the solutions are x = 2 or x = 3.
考虑方程 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0。令每个括号为零，得 x − 2 = 0 或 x − 3 = 0，因此解为 x = 2 或 x = 3。
x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3
After finding your solutions, always substitute them back into the original equation to verify. This catches sign errors and wrong factorisations quickly.
求出解之后，务必把它们代回原方程检验。这样可以快速发现符号错误或错误的因式分解。
4. Completing the Square | 配方法
Completing the square rewrites x² + bx as a perfect square plus a constant. The key identity is
配方法将 x² + bx 改写为一个完全平方加上一个常数。关键恒等式是
x² + bx = (x + b/2)² − (b/2)²
e, x² + 6x = (x + 3)² − 9. This method always works, even when factorisation is difficult.
例如，x² + 6x = (x + 3)² − 9。这种方法始终有效，即使因式分解困难时也能使用。
To solve x² + 6x + 1 = 0, first complete the square on x² + 6x:
要解 x² + 6x + 1 = 0，先对 x² + 6x 配方：
(x + 3)² − 9 + 1 = 0 → (x + 3)² = 8 → x + 3 = ±√8
Therefore x = −3 ± √8. In exact form this is x = −3 ± 2√2. You can leave it as √8 if the question asks for a surd, or round to a decimal if required.
因此 x = −3 ± √8。精确形式为 x = −3 ± 2√2。若题目要求保留根式形式则用 √8，若要求小数则可以四舍五入。
est way to find the vertex of a parabola, since y = (x − h)² + k has its vertex at (h, k).
配方法也是求抛物线顶点的最快方法，因为 y = (x − h)² + k 的顶点在 (h, k)。
5. The Quadratic Formula | 求根公式
any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula:
对于任意二次方程 ax² + bx + c = 0，其解由求根公式给出：
x = (−b ± √b² − 4ac) / 2a
This formula works for every quadratic equation, including those that cannot be factorised using integers. You should write down the values of a, b and c clearly before substituting.
这个公式适用于所有二次方程，包括那些无法用整数因式分解的方程。代入前应把 a、b、c 的值写清楚。
Worked example: solve 2x² − 5x − 3 = 0. Here a = 2, b = −5, c = −3. Substituting gives
例题：解 2x² − 5x − 3 = 0。这里 a = 2，b = −5，c = −3。代入得
x = (5 ± √25 + 24) / 4 = (5 ± √49) / 4 = (5 ± 7) / 4
1/2. Remember that b is negative in this example, so −b = 5, and b² = (−5)² = 25, not −25.
因此 x = (5 + 7)/4 = 3 或 x = (5 − 7)/4 = −1/2。注意本题中 b 为负数，所以 −b = 5，且 b² = (−5)² = 25，而不是 −25。
6. The Discriminant | 判别式
The expression inside the square root of the quadratic formula is called the discriminant, written as Δ:
求根公式中根号内的表达式称为判别式，记为 Δ：
Δ = b² − 4ac
The discriminant tells us how many real roots a quadratic equation has, without solving it fully. This is a frequent exam question.
判别式可以告诉我们二次方程有多少个实数根，而无需完整求解。这是常见的考点。
If Δ > 0, there are two distinct real roots.
若 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, there is one repeated real root.
若 Δ = 0，方程有一个重根（两个相等实数根）。
If Δ < 0, there are no real roots.
若 Δ < 0，方程没有实数根。
| Discriminant | 判别式 | Nature of roots | 根的性质 |
|---|---|
| Δ > 0 | Two distinct real roots | 两个不等实根 |
| Δ = 0 | One repeated root | 一个重根 |
| Δ < 0 | No real roots | 无实根 |
ere a = 1, b = 4, c = 5, so Δ = 16 − 20 = −4. Since Δ < 0, the equation has no real roots.
例题：判断 x² + 4x + 5 = 0 的根的性质。这里 a = 1，b = 4，c = 5，所以 Δ = 16 − 20 = −4。由于 Δ < 0，方程无实数根。
7. Graphs of Quadratic Functions | 二次函数的图象
ts, if they exist, are the roots of ax² + bx + c = 0. The y-intercept is the point (0, c).
y = ax² + bx + c 的图象是一条抛物线。若存在 x 轴交点，它们就是 ax² + bx + c = 0 的根。y 轴截距是点 (0, c)。
The axis of symmetry is the vertical line through the vertex. Its equation is
对称轴是经过顶点的竖直直线，其方程为
x = −b / 2a
3, the axis of symmetry is x = 4/2 = 2, and y = 4 − 8 + 3 = −1, so the vertex is (2, −1).
将该 x 值代回方程即可求得顶点的 y 坐标。例如，对于 y = x² − 4x + 3，对称轴为 x = 4/2 = 2，y = 4 − 8 + 3 = −1，因此顶点为 (2, −1)。
To sketch a quadratic graph, find the roots, the y-intercept, and the vertex, then draw a smooth curve through these points. Mark the axis of symmetry clearly.
画二次函数草图时，先求根、y 轴截距和顶点，然后过这些点画平滑曲线，并清楚标出对称轴。
8. Solving Quadratic Inequalities | 二次不等式
ite the related equation x² − 4x + 3 = 0 and solve: (x − 1)(x − 3) = 0, so x = 1 or x = 3.
像 x² − 4x + 3 > 0 这样的二次不等式，可以通过因式分解和画抛物线来解决。先写出对应方程 x² − 4x + 3 = 0 并求解：(x − 1)(x − 3) = 0，得 x = 1 或 x = 3。
e solution is x < 1 or x > 3. If the inequality were ≤ 0, the solution would be 1 ≤ x ≤ 3.
由于抛物线开口向上，图象在两个根之外位于 x 轴上方。因此解为 x < 1 或 x > 3。若不等式为 ≤ 0，则解为 1 ≤ x ≤ 3。
x² − 4x + 3 > 0 → x < 1 or x > 3
A sign table is another reliable method: test values in each interval and record whether the product is positive or negative. Use this to avoid mistakes when the quadratic does not factorise neatly.
符号表是另一种可靠方法：在每一区间内取值测试，记录乘积为正还是为负。当二次式不易分解时，用此方法可以避免出错。
9. Real-Life Applications | 实际应用
Many IGCSE problems translate a geometric or physical situation into a quadratic equation. A classic example is finding the length and width of a rectangle.
许多 IGCSE 应用题会把几何或物理情境转化为二次方程。一个经典例子是求矩形的长和宽。
ich expands to x² − 2x − 24 = 0. Factorising gives (x − 6)(x + 4) = 0, so x = 6 or x = −4.
一个矩形的长为 x cm，宽为 (x − 2) cm，面积为 24 cm²。方程为 x(x − 2) = 24，展开得 x² − 2x − 24 = 0。因式分解得 (x − 6)(x + 4) = 0，所以 x = 6 或 x = −4。
6 cm long and 4 cm wide. Always decide at the end which solutions are physically possible.
由于长度不能为负数，舍去 x = −4。该矩形长为 6 cm，宽为 4 cm。最后一定要判断哪些解在实际情况中合理。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Students frequently make the same errors in quadratic questions. Recognising these traps will save you marks on exam day.
学生在二次方程题目中经常犯同样的错误。识别这些陷阱能帮你在考场上拿到更多分数。
Forgetting to rearrange the equation to zero before factorising or using the formula.
忘记在因式分解或使用公式前将方程整理为零。
Sign errors when substituting negative values into the quadratic formula, especially b² and −b.
将负值代入求根公式时出现符号错误，尤其是 b² 和 −b。
Missing one of the two solutions when solving by taking square roots.
通过开平方求解时漏掉两个解中的一个。
Not checking answers by substituting back into the original equation.
没有把答案代回原方程进行检验。
Forgetting to reject negative or impossible answers in word problems.
在应用题中忘记舍去负数或不可能的答案。
Show your working clearly and write the quadratic formula in full before substituting. This earns method marks even if you make an arithmetic slip.
解题过程要写清楚，并在代入前完整写出求根公式。即使出现计算错误，也能获得方法分。
Finally, memorise the three key tools: factorisation, completing the square, and the quadratic formula. If a question says "show your working" or "solve algebraically", you must use one of these methods rather than just reading values from a graph.
最后，牢记三种核心工具：因式分解、配方法和求根公式。如果题目要求“写出过程”或“用代数方法求解”，就必须使用其中一种方法，而不能只从图象上读数。
With regular practice, solving quadratics becomes a routine skill. Use past papers to test yourself on all the formats covered in this guide, from simple factorisation to discriminant analysis and graph sketching.
通过定期练习，解二次方程会成为一项常规技能。用历年真题测试本指南涵盖的所有题型，从简单因式分解到判别式分析再到图象画法。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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