📚 054 Solving Quadratic Equations | 解二次方程
Quadratic equations form a core part of the IGCSE Edexcel Mathematics syllabus. Mastering solution methods—factorising, completing the square, and using the quadratic formula—is essential for success in both the Foundation and Higher tier papers. This guide breaks down the key techniques, common pitfalls, and exam strategies to help you tackle any quadratic confidently.
二次方程是 IGCSE Edexcel 数学大纲的核心内容。掌握因式分解、配方法和二次公式等解法对于在基础卷和进阶卷中取得成功至关重要。本指南将分解关键技巧、常见误区和考试策略，帮助你自信应对任何二次方程问题。
1. Understanding Quadratic Equations | 理解二次方程
A quadratic equation is any 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, which means there will usually be two solutions, or roots.
二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数且 a ≠ 0。x 的最高次幂为 2，因此通常会有两个解，即两个根。
For example, 2x² − 8x + 6 = 0 th a = 2, b = −8, c = 6. The solutions to this equation make the expression equal to zero.
例如，2x² − 8x + 6 = 0 是一个二次方程，其中 a = 2，b = −8，c = 6。方程的解使该表达式等于零。
2. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic expression can be written as a product of two linear brackets. For the equation x² − 5x + 6 = 0 ers that multiply to +6 and add to −5: these are −3 and −2. So we have (x − 3)(x − 2) = 0.
当二次表达式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。对于方程 x² − 5x + 6 = 0，寻找两个乘积为 +6 且和为 −5 的数字：它们是 −3 和 −2。因此得到 (x − 3)(x − 2) = 0。
s x = 2. Always check your solutions by substituting them back into the original equation.
令每个括号等于零：x − 3 = 0 得出 x = 3；x − 2 = 0 得出 x = 2。始终将解代入原方程进行验证。
3. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)² + q = 0, which is particularly useful when the quadratic does not factorise nicely. Take x² + 6x − 7 = 0 is 3, square it to get 9. Rewrite as (x² + 6x + 9) − 9 − 7 = 0, giving (x + 3)² − 16 = 0.
配方法将二次方程重写为 (x + p)² + q = 0 的形式，当二次方程难以因式分解时尤为有用。以 x² + 6x − 7 = 0 为例：x 项系数的一半是 3，将其平方得到 9。重写为 (x² + 6x + 9) − 9 − 7 = 0，即 (x + 3)² − 16 = 0。
= 1 and x = −7. This method also helps identify the vertex of the parabola when graphing.
然后得到 (x + 3)² = 16，所以 x + 3 = ±4。解得 x = 1 和 x = −7。此方法还有助于在绘图时确定抛物线的顶点。
4. Solving with the Quadratic Formula | 二次公式法
When a quadratic cannot be factorised easily, the quadratic formula provides a direct route: x = [−b ± √b² − 4ac] / 2a. This formula works for any quadratic in standard form.
当二次方程不易因式分解时，二次公式提供了一条直接的路径：x = [−b ± √b² − 4ac] / 2a。该公式适用于任何标准形式的二次方程。
For 3x² + 5x − 2 = 0, identify a = 3, b = 5, c = −2. Substitute into the formula: x = [−5 ± √25 + 24] / 6 = [−5 ± √49] / 6 = [−5 ± 7] / 6, giving x = 1/3 and x = −2.
对于 3x² + 5x − 2 = 0，确定 a = 3，b = 5，c = −2。代入公式：x = [−5 ± √25 + 24] / 6 = [−5 ± √49] / 6 = [−5 ± 7] / 6，得到 x = 1/3 和 x = −2。
- Always write the formula down before substituting to avoid sign errors.
- 代入前务必先写出公式，以避免符号错误。
5. The Discriminant | 判别式
The discriminant is the part under the square root: Δ = b² − 4ac. It tells you about the nature of the roots without solving the equation.
判别式是平方根号下的部分：Δ = b² − 4ac。它能在不解方程的情况下告诉你根的性质。
- If b² − 4ac > 0, there are two distinct real roots.
- 如果 b² − 4ac > 0，有两个不相等的实根。
- If b² − 4ac = 0, there is exactly one real root (a repeated root).
- 如果 b² − 4ac = 0，恰好有一个实根（重根）。
- If b² − 4ac < 0, there are no real roots (the graph does not cross the x-axis).
- 如果 b² − 4ac < 0，没有实根（图像不与 x 轴相交）。
Edexcel IGCSE questions often ask you to “find the value of k for which the equation has equal roots”. Set b² − 4ac = 0 and solve for k.
Edexcel IGCSE 试题经常要求“求使方程有等根的 k 值”。令 b² − 4ac = 0，然后求解 k。
6. Graphical Interpretation | 图像理解
opens downwards (a maximum point). The x-intercepts are the solutions to ax² + bx + c = 0.
二次函数 y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上（有最低点）；若 a < 0，开口向下（有最高点）。图像与 x 轴的交点即为方程 ax² + bx + c = 0 的解。
The vertex (turning point) is given by (−b/2a, f(−b/2a)). Completing the square makes the vertex clear: from (x + 3)² − 16, the vertex is (−3, −16).
顶点（转折点）坐标为 (−b/2a, f(−b/2a))。配方法可以使顶点一目了然：从 (x + 3)² − 16 可知顶点为 (−3, −16)。
7. Forming Quadratic Equations from Word Problems | 从实际问题建立二次方程
Many IGCSE problems ask you to form a quadratic from a real-life scenario, such as area, product of consecutive numbers, or projectile motion. Read carefully, assign a variable, and translate the conditions into an equation.
许多 IGCSE 题目要求从现实情境中建立二次方程，例如面积、连续数的乘积或抛体运动。仔细阅读题目，设定变量，并将条件转化为方程。
Example: “The product of two consecutive positive integers is 72.” . Factorising gives (n + 9)(n − 8) = 0, so n = 8 (positive root). The numbers are 8 and 9.
示例：“两个连续正整数的乘积为 72。”设两数为 n 和 n + 1。则 n(n + 1) = 72 → n² + n − 72 = 0。因式分解得 (n + 9)(n − 8) = 0，故 n = 8（正根）。两数为 8 和 9。
8. Difference of Two Squares | 平方差公式
r 4x² − 25 = 0, write it as (2x)² − 5² = (2x − 5)(2x + 5) = 0, giving x = 5/2 or x = −5/2.
特殊形式 x² − a² 可立即分解为 (x − a)(x + a)。识别出这一点可以节省时间。对于 4x² − 25 = 0，写为 (2x)² − 5² = (2x − 5)(2x + 5) = 0，解得 x = 5/2 或 x = −5/2。
es not factorise over the real numbers. It only has real roots if b² − 4ac ≥ 0 in context.
切勿将其与 x² + a² 混淆，后者在实数范围内不能因式分解。它在实数范围内有根的条件仍是 b² − 4ac ≥ 0。
9. Common Mistakes to Avoid | 常见错误
- Forgetting to set the equation to zero before factorising. Always rearrange to standard form first.
- 在因式分解前忘记将方程设为零。务必先整理成标准形式。
- Incorrectly applying the quadratic formula with a negative a value: watch signs, especially −b and −4ac.
- 在 a 为负数时错误应用二次公式：注意符号，尤其是 −b 和 −4ac。
- Dividing through by x without considering x = 0 as a root—this loses a solution.
- 不考虑 x = 0 根就直接除以 x——这会丢失一个解。
- Believing that a negative discriminant means the quadratic has no roots, rather than no real roots. Complex roots are beyond IGCSE.
- 认为负的判别式表示二次方程没有根，而实际上是没有实根。复数根超出了 IGCSE 的范围。
10. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧
When tackling a quadratic in an Edexcel IGCSE exam, first check if the equation can be factorised. If the coefficients are large or the question explicitly asks for answers to 2 decimal places, use the quadratic formula.
在 Edexcel IGCSE 考试中处理二次方程时，首先检查方程是否可以因式分解。如果系数较大或题目明确要求答案保留两位小数，则使用二次公式。
Present your work clearly: write down the formula, substitute carefully, and simplify step by step. For discriminant questions, state “two distinct real roots”, “one repeated root”, or “no real roots” in full sentences as required by the mark scheme.
清晰地展示解题过程：写下公式，仔细代入，并逐步化简。对于判别式问题，按评分标准的要求用完整语句写出“有两个不相等的实根”“有一个重根”或“没有实根”。
Use your calculator wisely. For completing the square, you can check your turning point using the vertex formula. And always re-read the question to ensure you have given the solutions in the requested form—as integers, fractions, or decimals.
明智地使用计算器。对于配方法，你可以用顶点公式验证转折点。此外，始终重读题目，确保你以所要求的形式（整数、分数或小数）给出答案。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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