📚 Topic 030: Solving Quadratic Equations | 专题030：解二次方程
re constants and a ≠ 0. These equations appear throughout algebra and are essential in many areas of mathematics, from geometry to physics. In the Edexcel IGCSE Mathematics syllabus, you are expected to master several methods for finding the solutions (also called roots) of quadratic equations, as well as understanding the role of the discriminant and solving real-life problems. This guide covers factorising, completing the square, the quadratic formula, graphical methods, quadratic inequalities, and how to tackle word problems.
二次方程是次数为二的多项式方程，一般形式为 ax² + bx + c = 0，其中 a, b, c 为常数且 a ≠ 0。这类方程在代数中随处可见，也是从几何到物理等许多数学领域的基础。在 Edexcel IGCSE 数学考试大纲中，你需要熟练掌握几种求二次方程解（也叫根）的方法，同时理解判别式的作用，并能解决实际应用题。本复习指南涵盖因式分解法、配方法、二次公式法、图像法、二次不等式以及如何应对文字题。
1. What is a Quadratic Equation? | 什么是二次方程？
ifying or expanding brackets they take on the standard form. For example, x(x + 5) = 6 is actually x² + 5x − 6 = 0. In IGCSE exams you will often be given equations that are already in standard form or can be easily rearranged.
二次方程是指能够变形为标准形式 ax² + bx + c = 0 的任何方程，其中 x 代表未知变量，a, b, c 为实数且 a ≠ 0。x 的最高次幂是 2，因此得名二次方程。有些方程起初看起来不像二次方程，但在化简或展开括号后就呈现标准形式。例如，x(x + 5) = 6 实际上是 x² + 5x − 6 = 0。在 IGCSE 考试中，给出的方程通常已经是标准形式，或者可以轻易变形为标准形式。
2. Solving by Factorising | 因式分解法求解
d c that combine to give b through cross multiplication. Once factorised as (px + q)(rx + s) = 0, set each bracket equal to zero to obtain the solutions x = −q/p or x = −s/r. Always check your solutions by substituting back into the original equation. This method is efficient but only works when the quadratic factorises nicely over the integers.
当二次式能写成两个一次括号的乘积时，因式分解通常是最快的方法。对于形如 x² + bx + c 的二次式，寻找两个数，使其乘积为 c，和为 b。如果 x² 的系数不是 1（即 a > 1），你需要考虑 a 和 c 的因数，通过十字相乘组合得到 b。一旦分解为 (px + q)(rx + s) = 0，令每个括号等于零即可得到解 x = −q/p 或 x = −s/r。务必通过代入原方程检查解的正确性。这种方法快速有效，但仅当二次式在整数范围内可分解时才适用。
- Worked example: ct = 10, sum = −7 → numbers −2 and −5 → factorises to (x − 2)(x − 5) = 0 → x = 2 or x = 5.
- 练习题：解方程 x² − 7x + 10 = 0。积为 10，和为 −7 → 找到 −2 和 −5 → 分解为 (x − 2)(x − 5) = 0 → x = 2 或 x = 5。
3. Solving by Completing the Square | 配方法求解
Completing the square transforms the quadratic expression into a perfect square plus or minus a constant. Starting with x² + bx, add (b/2)² to create (x + b/2)², then balance the equation. When a ≠ 1, first divide every term by a to obtain a leading coefficient of 1. This method is extremely useful because it works for any quadratic equation and leads directly to the vertex form of a parabola. It is also the foundation for deriving the quadratic formula. The steps: move the constant term to the other side, add (half of coefficient of x)² to both sides, factor the perfect square, then take square roots and solve for x.
配方法将二次式转化为完全平方加或减一个常数。从 x² + bx 开始，加上 (b/2)² 得到 (x + b/2)²，然后平衡方程。当 a ≠ 1 时，先将每一项除以 a，使首项系数为 1。此方法非常有用，因为它适用于任何二次方程，并能直接得到抛物线的顶点式。它也是推导二次公式的基础。步骤：将常数项移到等号另一边，两边同时加上（x 系数一半的平方），因式分解完全平方，然后开平方根并解出 x。
- Example: lve x² + 6x + 2 = 0. x² + 6x = −2. Add (6/2)² = 9 to both sides → (x + 3)² = 7 → x + 3 = ±√7 → x = −3 ± √7.
- 例子：解 x² + 6x + 2 = 0。x² + 6x = −2。两边加上 (6/2)² = 9 → (x + 3)² = 7 → x + 3 = ±√7 → x = −3 ± √7。
4. The Quadratic Formula | 二次公式
The quadratic formula gives the solutions of ax² + bx + c = 0 as:
x = (−b ± √b² − 4ac) / (2a)
This formula is derived from completing the square and works for every quadratic equation, even those with irrational or complex roots. To use it, simply identify the values of a, b and c from the standard form, substitute them carefully, and simplify. Pay special attention to signs; a common error is mishandling negative values of b or c. In the exam, always write down the formula before substituting numbers to gain method marks. The expression under the square root, b² − 4ac, is called the discriminant and determines the nature of the roots.
二次公式给出了 ax² + bx + c = 0 的解：
x = (−b ± √b² − 4ac) / (2a)
这个公式由配方法推导而来，适用于所有二次方程，即使是根为无理数或复数的情况。要使用它，只需从标准形式中确定 a, b, c 的值，仔细代入并化简。特别注意符号；常见错误是处理 b 或 c 为负值时出错。考试时，在代入数字之前先将公式写下来，这样可以获得方法步骤分。根号内的式子 b² − 4ac 称为判别式，决定根的性质。
- Example: 2x² − 4x − 3 = 0. a=2, b=−4, c=−3. x = [4 ± √16 + 24] / 4 = [4 ± √40] / 4 = [4 ± 2√10] / 4 = 1 ± (√10)/2.
- 例子：2x² − 4x − 3 = 0。a=2, b=−4, c=−3。x = [4 ± √16 + 24] / 4 = [4 ± √40] / 4 = [4 ± 2√10] / 4 = 1 ± (√10)/2。
5. The Discriminant and Nature of Roots | 判别式与根的性质
. If Δ = 0, there is exactly one real root (a repeated root, or the parabola touches the x‑axis). If Δ < 0, the quadratic has no real roots (the parabola does not intersect the x‑ al roots. You can also use the discriminant to check your answers when solving quadratics.
判别式 Δ = b² − 4ac 能告诉我们二次方程有多少个实数解，而无需完全解出方程。若 Δ > 0，则有两个不等的实根。若 Δ = 0，则恰好有一个实根（重根，或者说抛物线与 x 轴相切）。若 Δ < 0，则二次方程无实根（抛物线与 x 轴不相交）。在 IGCSE 考试中，这经常通过求交点个数或求使方程有等根的常数值来考查。你还可以用判别式来检验自己解二次方程时的答案。
| Δ = b² − 4ac | Nature of roots / 根的性质 |
| Δ > 0 | Two distinct real roots / 两个不等实根 |
| Δ = 0 | One real root (repeated) / 一个实根（重根） |
| Δ < 0 | No real roots / 无实根 |
6. Graphic Interpretation | 图像解释
h of y = ax² + bx + c is a parabola. The solutions of ax² + bx + c = 0 correspond to the x‑coordinates where the curve intersects the x‑axis. If the curve cuts the x‑axis at two points, the quadratic has two real distinct roots. If it just touches the x‑axis, there is one repeated root. If it never reaches the x‑ are no real roots. Knowing the shape is important: for a > 0 the parabola opens upwards (∪) and for a < 0 it opens downwards (∩ ions by reading off the intersections from a drawn graph, or to check algebraic solutions.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解对应于曲线与 x 轴交点的 x 坐标。如果曲线与 x 轴相交于两点，二次方程有两个不等实根；如果只接触 x 轴，则有一个重根；如果完全不到达 x 轴，则无实根。了解抛物线形状很重要：当 a > 0 时抛物线开口向上（∪），当 a < 0 时开口向下（∩）。图像法可用于通过绘制图形读取交点来解二次方程，也可用于检验代数解法。
4x + 3 = 0 graphically, plot y = x² − 4x + 3 and see where y = 0. The curve crosses the x‑axis at x = 1 and x = 3, so those are the solutions.
例如，用图像法解 x² − 4x + 3 = 0，画出 y = x² − 4x + 3 并找到 y = 0 的位置。该曲线在 x = 1 和 x = 3 处穿过 x 轴，因此解为 x = 1 和 x = 3。
7. Solving Quadratic Inequalities | 解二次不等式
od involves solving the corresponding quadratic equation to find critical values, then testing intervals on a number line. Always factorise or use the formula to find the roots, then sketch a rough parabola to determine where the expression is positive or negative. Remember that the sign of a influences whether the parabola opens up or down. Once you have the intervals, write the solution set using set notation or inequality signs. Be careful with strict inequalities versus non‑strict ones, and always consider whether the roots are included.
二次不等式的形式为 ax² + bx + c > 0 或 < 0 （或带有 ≥、≤）。解法是先解对应的二次方程以找到临界值，然后在数轴上测试各区间。总是通过因式分解或公式法求根，然后画出一条抛物线的草图，以判断表达式何时为正或为负。注意 a 的符号决定抛物线开口方向。得到区间后，使用集合符号或不等号写出解集。注意严格不等式和非严格不等式的区别，并始终考虑根是否包含在内。
- Example: 3. Parabola opens upwards. Expression is positive for x < 2 or x > 3. Solution: x < 2 or x > 3.
- 例子：解 x² − 5x + 6 > 0。根为 x = 2 和 x = 3。抛物线开口向上。当 x < 2 或 x > 3 时表达式为正。解为 x < 2 或 x > 3。
8. Word Problems Leading to Quadratic Equations | 应用题（化为二次方程）
Many real‑world situations give rise to quadratic equations, such as area problems, projectile motion, and number puzzles. To solve these, first define a variable, translate the given conditions into an algebraic equation, simplify it into the standard quadratic form, and then choose the appropriate method (factorising, formula, or completing the square). Always check whether all solutions make sense in the context of the problem; for instance, a negative length or time is usually discarded. Drawing a diagram and carefully reading the problem statement are essential for setting up the equation correctly.
许多实际情境会产生二次方程，例如面积问题、抛体运动以及数字谜题。解决这类问题，首先要设出变量，将所给条件转化为代数方程，化简成标准二次形式，然后选择合适的方法（因式分解、公式法或配方法）。务必检查所有解在问题背景下是否有意义；例如，负的长度或时间通常舍去。绘制示意图并仔细阅读题目描述，对于正确建立方程至关重要。
- Typical problem: 72 → n² + n − 72 = 0 → (n+9)(n−8)=0 → n=8 or n=−9. Discard −9, so the numbers are 8 and 9.
- 典型问题：两个连续正整数的积为 72。求这两个数。设这两个数为 n 和 n+1，则 n(n+1) = 72 → n² + n − 72 = 0 → (n+9)(n−8)=0 → n=8 或 n=−9。舍去 −9，因此两数为 8 和 9。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
first. Mistake 2: Losing a sign when substituting into the quadratic formula, especially with negative b values. Write the formula and substitute in brackets to keep signs clear. Mistake 3: Incorrectly completing the square when a ≠ 1. Divide every term by a first. Mistake 4: Assuming a quadratic always has two real roots without checking the discriminant. Mistake 5: Writing the final roots without simplifying surds. Leave answers in exact surd form unless instructed otherwise. Mistake 6: Ignoring context in word problems – always verify that roots fit the physical situation.
错误 1：在因式分解前忘记将方程设为零。务必先将方程变形为 ax² + bx + c = 0 的形式。错误 2：代入二次公式时丢掉正负号，尤其是 b 为负值时。先写出公式并用括号代入数字，以保持符号清晰。错误 3：当 a ≠ 1 时配方法使用错误。每项先除以 a。错误 4：不检查判别式就默认二次方程总有两个实根。错误 5：写出最终根时未化简根式。除非题目要求，否则用准确根式形式保留答案。错误 6：在应用题中忽略实际背景——务必检验根是否符合实际情况。
10. Summary and Key Points | 总结与要点
hen possible; confidently complete the square; use the quadratic formula accurately; interpret the discriminant; relate algebraic solutions to a parabola' one for each question. With thorough practice, quadratics become a predictable and highly scoring topic.
要在 Edexcel IGCSE 数学中成功解决二次方程问题，请确保你能够：识别标准形式 ax² + bx + c = 0；在可能时用因式分解求解；熟练地进行配方；准确使用二次公式；解释判别式；将代数解与抛物线图像联系起来；求解二次不等式并将知识应用于实际问题。练习混合使用所有方法，以便针对每一道题选择最高效的方法。通过充分的练习，二次方程会成为一个可预测且得分率很高的专题。
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