📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are a fundamental topic in the Edexcel IGCSE Mathematics syllabus. Mastering them requires understanding their structure, applying different solution methods, and interpreting results in context.
二次方程是 Edexcel IGCSE 数学课程中的一个基础主题。掌握它需要理解其结构、运用不同的求解方法，并结合实际情境解读结果。
1. What is a Quadratic Equation? | 什么是二次方程
A quadratic equation is a polynomial equation of degree 2. Its general form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
二次方程是次数为 2 的多项式方程。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。
The highest power of the variable x is 2, which gives the equation its name ‘quadratic’.
变量 x 的最高次幂为 2，二次方程因此得名。
2. Standard Form and Key Terms | 标准形式与关键术语
In the standard form ax² + bx + c = 0, a is called the leading coefficient, b is the linear coefficient, and c is the constant term.
在标准形式 ax² + bx + c = 0 中，a 称为二次项系数，b 称为一次项系数，c 称为常数项。
A quadratic equation may have zero, one, or two real solutions, depending on the value of its discriminant.
根据判别式的值，二次方程可能有零个、一个或两个实数解。
3. Solving by Factorisation | 因式分解法
Factorisation involves writing the quadratic as a product of two linear brackets. For example, x² − 5x + 6 = 0 can be written as (x − 2)(x − 3) = 0.
因式分解法是将二次式写成两个一次括号的乘积。例如，x² − 5x + 6 = 0 可以写成 (x − 2)(x − 3) = 0。
Then, using the zero product property, either x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.
然后利用零乘积性质，x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。
This method works best when the equation has simple integer factors.
当方程有简单的整数因式时，这种方法最有效。
4. Solving by Completing the Square | 配方法
Completing the square rewrites the equation in the form a(x + p)² + q = 0. For x² + 6x + 5 = 0, we add and subtract (6/2)² = 9.
配方法将方程改写为 a(x + p)² + q = 0 的形式。对于 x² + 6x + 5 = 0，我们加上并减去 (6/2)² = 9。
Thus x² + 6x + 9 − 9 + 5 = 0, giving (x + 3)² − 4 = 0.
因此 x² + 6x + 9 − 9 + 5 = 0，得到 (x + 3)² − 4 = 0。
Then (x + 3)² = 4, so x + 3 = ± 2, leading to x = −1 or x = −5.
然后 (x + 3)² = 4，所以 x + 3 = ± 2，得出 x = −1 或 x = −5。
This method is particularly useful when factoring is difficult and for deriving the quadratic formula.
当因式分解困难时，配方法尤其有用，它也用于推导二次公式。
5. The Quadratic Formula | 二次公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0 directly. The solutions are given by:
二次公式可以直接解任何二次方程 ax² + bx + c = 0。解为：
x = (−b ± √(b² − 4ac)) / (2a)
This formula is derived by completing the square on the general form.
该公式通过对一般形式配方推导得出。
It works for all cases, including when the solutions are irrational or non-real.
它适用于所有情况，包括解为无理数或无实数解的情形。
6. The Discriminant | 判别式
The discriminant, denoted Δ, is the part of the formula under the square root: Δ = b² − 4ac.
判别式用 Δ 表示，是公式中根号内的部分：Δ = b² − 4ac。
Its value tells us the nature of the roots without solving the equation:
它的值无需解方程即可告诉我们根的性质：
| Δ value / Δ的值 | Nature of roots / 根的性质 |
| Δ > 0 | Two distinct real roots / 两个不同的实数根 |
| Δ = 0 | One repeated real root / 一个重实数根 |
| Δ < 0 | No real roots / 没有实数根 |
If Δ is a perfect square, the roots are rational and factorisation is possible.
如果 Δ 是完全平方数，则根为有理数，可以用因式分解求解。
7. Graphical Representation | 图像表示
The graph of a quadratic function y = ax² + bx + c is a parabola. The roots are the x-coordinates where the parabola crosses the x-axis.
二次函数 y = ax² + bx + c 的图像是一条抛物线。根是抛物线与 x 轴交点的 x 坐标。
If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
若 a > 0，抛物线开口向上；若 a < 0，抛物线开口向下。
The vertex and line of symmetry can also be found from the completed square form.
顶点和对称轴也可以通过配方法得到。
8. Solving Quadratic Inequalities | 解二次不等式
Quadratic inequalities involve signs such as , ≤, or ≥. For example, x² − 4 > 0.
二次不等式涉及 、≤ 或 ≥ 等符号。例如，x² − 4 > 0。
First, solve the corresponding quadratic equation to find critical points. For x² − 4 = 0, x = −2 or x = 2.
首先解对应的二次方程以找到临界点。对于 x² − 4 = 0，x = −2 或 x = 2。
Then test intervals to determine where the inequality holds. The solution is x < −2 or x > 2.
然后测试区间以确定不等式成立的范围。解为 x < −2 或 x > 2。
A graph or sign table is often helpful for solving such inequalities.
画图或使用符号表通常有助于解这类不等式。
9. Applications in Problem Solving | 实际应用
Quadratic equations appear in geometry, physics, and economics. For example, finding the dimensions of a rectangle with a given area and perimeter.
二次方程出现在几何、物理和经济学中。例如，求给定面积和周长的矩形尺寸。
Projectile motion problems use quadratics to model height over time. The equation s = ut + ½at² is quadratic in time t.
抛体运动问题使用二次方程来模拟高度与时间的关系。方程 s = ut + ½at² 关于时间 t 是二次的。
You must often form an equation from a word problem, then solve it effectively.
通常你需要从实际问题中建立方程，然后有效地求解。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common errors include forgetting to set the equation to zero before factoring, misapplying the quadratic formula, and ignoring negative roots.
常见错误包括：因式分解前忘记将方程化为零、错误套用二次公式，以及忽略负根。
Always check your solutions by substituting back into the original equation.
始终通过代入原方程来检验你的解。
In exams, show all steps clearly. The quadratic formula can save time when factoring is not obvious.
考试中，清晰地写出所有步骤。当因式分解不明显时，二次公式可以节省时间。
Remember to write your final answers to the required degree of accuracy, often 2 decimal places or 3 significant figures.
记住按要求的精度写出最终答案，通常是 2 位小数或 3 位有效数字。
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