📚 Solving Quadratic Equations | 解二次方程
Quadratic equations lie at the very heart of IGCSE Edexcel Mathematics. Whether factorising simple trinomials, completing the square or applying the quadratic formula, a confident command of these methods will unlock a large portion of the algebra and graph-sketching questions you will face in the exam. This article breaks down every essential technique, shows you how the discriminant decides the nature of roots, and explains how to handle quadratic inequalities – all with worked examples that match the style and depth required by the Edexcel specification.
二次方程是 IGCSE Edexcel 数学的核心内容。不论是因式分解简单的三项式、配方法还是使用求根公式，熟练掌握这些方法将帮助你解决考试中大量的代数与作图题。本文逐一拆解每一种关键方法，展示判别式如何决定根的性质，并说明如何处理二次不等式——每个步骤都配有符合 Edexcel 考纲风格与深度要求的例题。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation in one variable is an equation that can be written in the standard form ax² + bx + c = 0 here a, b and c are constants and a ≠ 0. If a = 0 the equation becomes linear, so the word 'quadratic' aph y = ax² + bx + c is a parabola that opens upwards when a > 0 and downwards when a < 0.
一元二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数且 a ≠ 0。如果 a = 0 方程就变成了一次方程，因此“二次”始终意味着 x² 项存在。图像 y = ax² + bx + c 的形状是一条抛物线，当 a > 0 时开口向上，当 a < 0 时开口向下。
, find the turning point of a quadratic graph, or interpret the roots on a sketch. Mastering the link between the algebraic form and the graph saves time and helps avoid sign errors.
IGCSE Edexcel 考题常要求解出类似 2x² - 5x - 3 = 0 的方程、找出二次图像的最高或最低点，或在草图上解释根的含义。掌握代数式与图像之间的联系能够节省时间并避免符号错误。
2. Solving by Factorising | 因式分解法求解
le quadratic where a = 1, you only need two numbers whose product is c and whose sum is b.
当二次式可以整齐地用整数因式分解时，因式分解法是最快捷的方法。将方程写成 ax² + bx + c = 0 的形式，找出两个括号 (px + q)(rx + s)，使它们相乘得到原式，然后令每个括号等于零并解出 x。对于 a = 1 的简单二次式，只需找到两个数，它们的乘积为 c、和为 b。
Example: (x - 5) = 0, giving x = 2 or x = 5. Always check by substituting back. When a ≠ 1, use the 'ac' method or systematic trial of factors.
例题：解 x² - 7x + 10 = 0。数字 -2 和 -5 乘积为 +10、和为 -7，因此 (x - 2)(x - 5) = 0，解得 x = 2 或 x = 5。务必代回检验。当 a ≠ 1 时，可使用“ac”方法或有条理地尝试分解因数。
3. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This form instantly reveals the turning point (–p, q) of the parabola and is the foundation for deriving the quadratic formula. Start by taking out any factor of a from the x² and x terms, then add and subtract (b/2a)² inside the bracket.
配方法把 ax² + bx + c 重写为 a(x + p)² + q 的形式。这种形式能立刻显示抛物线的顶点 (–p, q)，也是推导求根公式的基础。首先将 x² 项和 x 项中的 a 提取出来，然后在括号内加上并减去 (b/2a)²。
Example: in the form a(x + p)² + q. Factor 2: 2[x² + 4x] + 5. Half of 4 is 2, 2² = 4, so 2[(x + 2)² – 4] + 5 = 2(x + 2)² – 8 + 5 = 2(x + 2)² – 3. The turning point is (–2, – n, set the completed square equal to zero, isolate the squared term and take square roots.
例题：将 2x² + 8x + 5 表示为 a(x + p)² + q 的形式。提取 2：2[x² + 4x] + 5。4 的一半是 2，2² = 4，于是 2[(x + 2)² – 4] + 5 = 2(x + 2)² – 8 + 5 = 2(x + 2)² – 3。顶点为 (–2, –3)。解方程时，令配成的完全平方等于零，分离平方项再开方即可。
4. The Quadratic Formula | 求根公式
use the quadratic formula. For any quadratic ax² + bx + c = 0, the solutions are given by:
当因式分解困难或根为无理数时，使用求根公式。对于任意二次方程 ax² + bx + c = 0，解由下式给出：
x = [ –b ± √b² – 4ac ] / (2a)
You must learn this formula exactly. Pay close attention when b is negative; squaring a negative gives a positive, but –b will become the opposite sign. Always simplify the square root if possible and divide both terms in the numerator by the denominator separately if needed.
你必须准确记住这个公式。当 b 为负数时要格外小心；负数平方得正，但 –b 会变成相反符号。若可能，化简平方根，必要时将分子两项分别除以分母。
Example: Solve 3x² - 5x - 2 = 0. Here a = 3, b = –5, c = –2. Substituting: x = [5 ± √( (–5)² – 4×3×(–2) )] / (2×3) = [5 ± √25 + 24] / 6 = [5 ± √49] / 6 = (5 ± 7)/6. So x = 12/6 = 2 or x = –2/6 = –1/3. Both roots are rational because the discriminant is a perfect square.
例题：解 3x² - 5x - 2 = 0。这里 a = 3，b = –5，c = –2。代入：x = [5 ± √( (–5)² – 4×3×(–2) )] / (2×3) = [5 ± √25 + 24] / 6 = [5 ± √49] / 6 = (5 ± 7)/6。因此 x = 12/6 = 2 或 x = –2/6 = –1/3。由于判别式为完全平方数，两个根都是有理数。
5. Discriminant and Nature of Roots | 判别式与根的性质
The expression under the square root, Δ = b² – (a repeated root); if Δ < 0 there are no real roots (the curve does not cross the x-axis).
平方根号内的表达式 Δ = b² – 4ac 称为判别式。它能告诉你根的性质而不必完全解出方程：若 Δ > 0，有两个相异实根；若 Δ = 0，有唯一实根（重根）；若 Δ < 0，无实根（曲线不与 x 轴相交）。
Edexcel often asks: 'Find the value of k for which the equation has equal roots.' d solve for k. This tests your ability to link algebraic conditions and parameter changes.
Edexcel 常会这样考：“求使得方程有等根的 k 值。”令 Δ = 0 并解出 k。这考查你将代数条件与参数变化联系起来的能力。
Example: For x² + (k + 2)x + 9 = 0 to have equal roots, discriminant must be zero: (k + 2)² – 4×1×9 = 0 → (k + 2)² = 36 → k + 2 = ±6 → k = 4 or k = –8.
例题：要使 x² + (k + 2)x + 9 = 0 有等根，判别式必须为零：(k + 2)² – 4×1×9 = 0 → (k + 2)² = 36 → k + 2 = ±6 → k = 4 或 k = –8。
6. Graphical Interpretation of Solutions | 解的图像意义
f ax² + bx + c = 0 are the x-coordinates where the graph of y = ax² + bx + c crosses the x‑axis. The vertex (turning point) lies midway between the roots, at x = –b/(2a). If the graph does not intersect the x-axis, there are no real solutions.
方程 ax² + bx + c = 0 的解就是图像 y = ax² + bx + c 与 x 轴交点的横坐标。顶点位于两个根的正中间，其 x 坐标为 –b/(2a)。若图像不与 x 轴相交，则没有实数解。
rcept (0, c), the roots if they exist, and the turning point. Use the line of symmetry x = – ched graph and ask you to estimate solutions or solve a related equation such as f(x) = k.
作图时，先找出 y 轴截距 (0, c)、存在的根以及顶点。利用对称轴 x = –b/(2a) 检验图像形状。IGCSE 题目可能给出一个草图，要求你估算解或解出相关方程如 f(x) = k。
7. Solving Quadratic Inequalities | 解二次不等式
To solve ax² + bx + c < 0 or > 0, first treat it as an equation and find the roots. These roots divide the number line into intervals. Choose a test point from each interval to determine where the inequality holds. Alternatively, sketch the parabola and read off where it lies above or below the x-axis.
要解 ax² + bx + c < 0 或 > 0 的二次不等式，首先把它当作方程来求出根。这些根将数轴分成几个区间。从每个区间选取测试点来判断不等式是否成立。也可以先画出抛物线，观察图像在哪部分位于 x 轴上方或下方。
Critical rule: When the quadratic is positive and you require it to be less than zero, the solution lies between the roots; when greater than zero, it lies outside the roots. However, always check the sign of 'a' because a negative 'a' flips the logic.
关键规则：若 a > 0 且要求 ax² + bx + c < 0，解在两根之间；若要求 > 0，解在两根之外。但务必先检查 a 的符号，因为 a 为负时会反转逻辑。
Example: Solve x² - 5x + 6 < 0. Roots are 2 and 3, a > 0, so solution is 2 < x < 3.
例题：解 x² - 5x + 6 < 0。根为 2 和 3，a > 0，因此解为 2 < x < 3。
8. Applications and Word Problems | 应用题与文字题
Quadratic equations model many real-life scenarios: area problems, projectile motion, profit optimisation and number puzzles. Translate the words into an equation, solve, and then judge whether the answers make sense in the original context. Often one root will be rejected because it is negative or too large.
二次方程可以模拟许多实际问题：面积问题、抛体运动、利润最优化和数字谜题。先将文字转化为方程，求解，然后判断答案在原情境中是否有意义。通常一个根会因为负数或过大而被舍去。
Be systematic: assign a variable, write the relationship as an algebraic equation, reduce it to standard quadratic form, solve using the most efficient method, and state the final answer clearly with units if required. Edexcel expects you to show all steps and interpret your result briefly.
要有条理：设定变量，将关系写成代数方程，整理成标准二次形式，用最有效的方法求解，然后清晰给出最终答案（如有需要附上单位）。Edexcel 要求你展示所有步骤并简要解释结果。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
- Forgetting to set the equation to zero before factorising. Always rearrange so one side is 0.
- Losing the sign of b when using the quadratic formula. If b = –4, then –b = +4.
- rrectly when the numerator contains a ± term. Split into two separate fractions if unsure.
- Forgetting that (x + p)² = q has two solutions: x + p = ±√q, not just √q.
- Misidentifying a, b and c when the equation is not in standard order. Write it in descending powers of x first.
- 因式分解前忘记将方程整理为等于零的形式。务必先移项使一边为零。
- 使用求根公式时弄错 b 的符号。如果 b = –4，那么 –b 就是 +4。
- 分子中含有 ± 时通分错误。若不肯定，可拆成两个分数。
- 忘记 (x + p)² = q 有两个解： x + p = ±√q，而不只是 √q。
- 方程未按 x 的降幂排列时认错 a、b、c。先写成标准顺序。
10. Exam Strategy and Practice Tips | 考试策略与练习建议
ng the square can be neat. Otherwise, the quadratic formula works for every case and is safe when you are short on time.
先判断哪种方法适合给定的二次式。如果可以整数因式分解，因式分解最快。如果 a = 1 且 b 为偶数，配方法也很简洁。除此之外，求根公式适用于所有情况，时间紧张时也十分可靠。
Show all working; Edexcel awards method marks even if the final answer is wrong. Always check your roots by substitution or by comparing with a quick sketch. Practise mixed exercises so you can switch between methods fluently, and time yourself on past-paper questions to build speed and confidence.
展示全部解题步骤；即使最终答案错误，Edexcel 也会给方法分。始终通过代回检验或与简单草图对比来检查根的正确性。多做混合练习以在不同方法间熟练切换，并用往年真题计时训练，提高速度与信心。
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