📚 Mastering Quadratic Equations: Factorising, Formula and Graphs | 掌握二次方程：因式分解、公式法与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to geometry and word problems. Understanding how to solve them using multiple methods, and how their graphs behave, is essential for success in both Core and Extended papers. This revision guide covers the standard form of a quadratic, three algebraic solution methods, the discriminant, graphical interpretation, and common exam techniques.
二次方程贯穿 IGCSE 数学考试的各个部分，从代数、图像到几何和应用题。掌握多种解法并理解其图像特征，对于在核心卷和扩展卷中取得好成绩至关重要。本文复习二次方程的标准形式、三种代数解法、判别式、图像解读以及常见考试技巧。
1. What is a Quadratic Equation? | 什么是二次方程？
e constants and a ≠ 0. The highest power of the variable x is 2, which is why it is called 'quadratic' (from the Latin word for square). If a = 0, the equation becomes linear, not quadratic.
二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。未知数 x 的最高次数为 2，因此称为“二次”（源自拉丁语“平方”）。如果 a = 0，方程就变成一次方程，不再是二次方程。
o real solutions, one repeated real solution, or no real solutions, depending on the value of the discriminant, which we will explore later.
二次方程的例子包括 x² - 5x + 6 = 0、2x² + 3x - 2 = 0 和 x² - 9 = 0。二次方程可能有两个实数解、一个重复实数解或没有实数解，具体取决于判别式的值，我们将在后面探讨。
2. Standard Form and Key Terms | 标准形式与关键术语
nd c is the constant term. For example, in 3x² - 7x + 2 = 0, we have a = 3, b = -7, c = 2.
二次方程的标准形式是 ax² + bx + c = 0。其中 a 是 x² 的系数，称为首项系数；b 是 x 的系数；c 是常数项。例如，在 3x² - 7x + 2 = 0 中，a = 3，b = -7，c = 2。
equals sign, and like terms combined. For instance, 2x² = 5x - 3 becomes 2x² - 5x + 3 = 0.
在应用任何解法之前，必须先把方程整理成标准形式。可能需要移项并合并同类项。例如，2x² = 5x - 3 可整理为 2x² - 5x + 3 = 0。
- a must not be zero, otherwise the equation is not quadratic.
- a 不能为零，否则方程就不是二次方程。
- b and c can be zero, giving special forms such as ax² + c = 0 or ax² + bx = 0.
- b 和 c 可以为零，从而得到特殊形式，如 ax² + c = 0 或 ax² + bx = 0。
3. Solving by Factorising | 因式分解法求解
s) = 0, then use the fact that if a product equals zero, at least one factor must be zero.
当二次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解法。要用因式分解法解 ax² + bx + c = 0，首先把左边写成 (px + q)(rx + s) = 0，然后利用“乘积为零则至少一个因式为零”的性质。
2)(x - 3). Setting each factor to zero gives x - 2 = 0 or x - 3 = 0, hence x = 2 or x = 3.
例如，解 x² - 5x + 6 = 0。我们寻找两个数，它们的乘积为 +6，和为 -5。这两个数是 -2 和 -3，因此 x² - 5x + 6 = (x - 2)(x - 3)。令每个因式等于零，得到 x - 2 = 0 或 x - 3 = 0，所以 x = 2 或 x = 3。
= 2x² + 6x + x + 3 = 2x² + 7x + 3. Then set each factor to zero to find x = -½ or x = -3.
当首项系数 a 不为 1 时，因式分解需要更加小心。例如，2x² + 7x + 3 可以分解为 (2x + 1)(x + 3)。你可以展开验证：(2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3。然后令每个因式为零，解得 x = -½ 或 x = -3。
4. Completing the Square | 配方法
Completing the square transforms a quadratic expression into the form a(x + p)² + q. This method is particularly useful when the quadratic does not factorise neatly, and it also helps in finding the vertex of a quadratic graph. The key idea is to create a perfect square trinomial from the x² and x terms.
配方法将二次式转化为 a(x + p)² + q 的形式。当二次式不容易因式分解时，这种方法特别有用，同时它还能帮助确定二次函数图像的顶点。关键思想是利用 x² 项和 x 项构造完全平方三项式。
= 0 becomes (x + 3)² - 4 = 0, so (x + 3)² = 4, giving x + 3 = ±2, hence x = -1 or x = -5.
要对 x² + bx 进行配方，需要加上并减去 (b/2)²。例如，x² + 6x + 5 可以写成 x² + 6x + 9 - 9 + 5 = (x + 3)² - 4。因此方程 x² + 6x + 5 = 0 变为 (x + 3)² - 4 = 0，即 (x + 3)² = 4，得到 x + 3 = ±2，所以 x = -1 或 x = -5。
, 2x² - 8x + 3 = 2(x² - 4x) + 3 = 2[(x - 2)² - 4] + 3 = 2(x - 2)² - 8 + 3 = 2(x - 2)² - 5.
如果首项系数不是 1，应先将它从 x² 项和 x 项中提出来。例如，2x² - 8x + 3 = 2(x² - 4x) + 3 = 2[(x - 2)² - 4] + 3 = 2(x - 2)² - 8 + 3 = 2(x - 2)² - 5。
5. The Quadratic Formula | 求根公式
des a direct way to solve any quadratic equation ax² + bx + c = 0. The roots are given by:
求根公式提供了求解任何二次方程 ax² + bx + c = 0 的直接方法。其根由以下公式给出：
x = (-b ± √b² - 4ac) / 2a
This formula is derived from completing the square on the general quadratic equation. It is extremely useful when factorising is difficult or when the coefficients are decimals or fractions. Always substitute the values of a, b and c carefully, paying attention to signs.
这个公式是通过对一般二次方程进行配方推导出来的。当因式分解困难或系数为小数、分数时，它非常有用。代入 a、b、c 的值时务必仔细，注意正负号。
a = 3, b = 5, c = -2. Substituting gives x = (-5 ± √(5² - 4 × 3 × (-2))) / (2 × 3) = (-5 ± √25 + 24) / 6 = (-5 ± √49) / 6 = (-5 ± 7) / 6. Thus x = 2/6 = 1/3 or x = -12/6 = -2.
示例：用求根公式解 3x² + 5x - 2 = 0。这里 a = 3，b = 5，c = -2。代入得 x = (-5 ± √(5² - 4 × 3 × (-2))) / (2 × 3) = (-5 ± √25 + 24) / 6 = (-5 ± √49) / 6 = (-5 ± 7) / 6。因此 x = 2/6 = 1/3 或 x = -12/6 = -2。
When quoting the formula in an exam, show the substitution step clearly to gain method marks, even if the final answer is wrong due to arithmetic slips.
在考试中使用该公式时，要清晰展示代入步骤，即使最终答案因计算错误而出错，也能获得方法分。
6. The Discriminant | 判别式
The expression under the square root in the quadratic formula, b² - 4ac, is called the discriminant. It tells you how many real solutions a quadratic equation has:
求根公式中平方根下的表达式 b² - 4ac 称为判别式。它能告诉你二次方程有多少个实数解：
- If b² - 4ac > 0, the equation has two distinct real roots.
- 如果 b² - 4ac > 0，方程有两个不同的实数根。
- If b² - 4ac = 0, the equation has one repeated real root (or two equal roots).
- 如果 b² - 4ac = 0，方程有一个重复实数根（或两个相等根）。
- If b² - 4ac < 0, the equation has no real roots (the graph does not cross the x-axis).
- 如果 b² - 4ac < 0，方程没有实数根（图像不与 x 轴相交）。
nd the value of k for which a quadratic has equal roots. Set b² - 4ac = 0 and solve for k.
例如，对于 x² - 4x + 4 = 0，判别式为 (-4)² - 4 × 1 × 4 = 16 - 16 = 0，因此方程在 x = 2 处有一个重复根。考试中可能会要求你求出使二次方程有相等根的 k 值，这时令 b² - 4ac = 0 并解出 k。
7. Graphs of Quadratic Functions | 二次函数图像
a graph called a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The turning point is also called the vertex.
二次函数 y = ax² + bx + c 的图像称为抛物线。如果 a > 0，抛物线开口向上，有一个最低点；如果 a < 0，抛物线开口向下，有一个最高点。这个转折点也称为顶点。
= -b / (2a), and the y-coordinate is found by substituting this x back into the function.
顶点可以通过配方法求得。对于 y = (x + p)² + q，顶点坐标为 (-p, q)。或者，顶点的 x 坐标由 x = -b / (2a) 给出，y 坐标通过将该 x 值代回函数求得。
The axis of symmetry is the vertical line x = -b / (2a), which passes through the vertex.
图像与 y 轴的交点为 (0, c)。如果存在 x 轴交点，它们就是方程 ax² + bx + c = 0 的解。对称轴是经过顶点的垂直线 x = -b / (2a)。
8. Using Graphs to Solve Equations | 利用图像解方程
lve an equation such as x² - 3x - 4 = 0 by reading off where the curve crosses the x-axis.
二次函数图像可以用来解相关方程。y = ax² + bx + c 的 x 轴截距就是方程 ax² + bx + c = 0 的解。考试题目经常要求你画出图像，然后通过读取曲线与 x 轴的交点来解方程，如 x² - 3x - 4 = 0。
raph of y = x² - 3x - 4. First rearrange the target equation so one side matches the graph' ewritten as x² - 3x - 4 = -2, so the solutions are the x-values where y = -2 on the graph.
你可能还会被要求利用 y = x² - 3x - 4 的图像来解方程 x² - 3x - 2 = 0。首先将目标方程变形，使一边与图像表达式相同。这里 x² - 3x - 2 = 0 可改写为 x² - 3x - 4 = -2，因此解就是图像上 y = -2 时对应的 x 值。
When drawing a quadratic graph, always plot at least five points, including the vertex, the y-intercept, and symmetric points on either side of the axis of symmetry. Use a smooth curve, not straight line segments.
画二次函数图像时，至少标出五个点，包括顶点、y 轴截距以及对称轴两侧的对称点。用平滑曲线连接，不要用直线段。
9. Word Problems Involving Quadratics | 二次方程应用题
Many IGCSE problems involve forming a quadratic equation from a real-life situation. Common contexts include area of rectangles, projectile motion, product of consecutive numbers, and profit or cost problems. The key is to define a variable, write an equation, and then solve it using one of the methods above.
许多 IGCSE 题目要求根据实际情境列出二次方程。常见情境包括矩形面积、抛体运动、连续数的乘积以及利润或成本问题。关键在于设未知数、列出方程，然后用上述方法求解。
- 5) = 0, so x = -8 or x = 5. Since width cannot be negative, x = 5 cm and length = 8 cm.
示例：一个矩形的长比宽多 3 cm，面积为 40 cm²，求其尺寸。设宽为 x cm，则长为 x + 3 cm。面积 = x(x + 3) = 40，得到 x² + 3x - 40 = 0。因式分解得 (x + 8)(x - 5) = 0，所以 x = -8 或 x = 5。由于宽度不能为负，x = 5 cm，长为 8 cm。
Always check that your answers make sense in the context of the problem. Reject negative lengths, times or quantities unless the situation allows them.
一定要检查答案在题目情境中是否合理。除非情境允许，否则要舍去负的长度、时间或数量。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Students often lose marks due to sign errors, forgetting to set the equation to zero before factorising, or misquoting the quadratic formula. Another common mistake is dividing both sides of an equation by x without considering that x might be zero, which loses a solution.
学生常因符号错误、在因式分解前忘记将方程化为零、或记错求根公式而丢分。另一个常见错误是两边同除以 x 而没有考虑 x 可能为零，从而丢失一个解。
When solving by factorising, always check that the product of the constant terms equals c and the sum of the inner and outer terms equals b. For the quadratic formula, write down the formula first, then substitute values inside brackets to avoid sign mistakes.
用因式分解法求解时，一定要检查常数项乘积是否等于 c，以及内外项之和是否等于 b。使用求根公式时，先写下公式，再将数值代入括号内，以避免符号错误。
In graph questions, label your axes and use a sharp pencil. If the question asks for solutions to a given quadratic, state both the x-values clearly. If the discriminant is negative, write 'no real roots' rather than attempting to take the square root of a negative number.
在图像题中，标注坐标轴并使用削尖的铅笔。如果题目要求给出某个二次方程的解，要清晰地写出两个 x 值。如果判别式为负，应写“无实数根”，而不是尝试对负数开平方。
11. Summary and Key Formulas | 总结与关键公式
Quadratic equations are a central topic in IGCSE Mathematics. You should be confident in factorising, completing the square, and using the quadratic formula. Understanding the discriminant helps you predict the nature of roots, while graphing skills allow you to visualise solutions and solve related equations.
二次方程是 IGCSE 数学的核心主题。你应当熟练掌握因式分解、配方法和求根公式。理解判别式有助于预判根的性质，而图像技能使你能够直观地看到解并解决相关方程。
| Method | Formula / Procedure | When to Use |
|---|---|---|
| Factorising | (px + q)(rx + s) = 0 | When factors are easy to find |
| Completing the square | a(x + p)² + q = 0 | For vertex form or non-factorable quadratics |
| Quadratic formula | x = (-b ± √b² - 4ac) / 2a | Always works, especially for decimals or messy coefficients |
(2a), and the axis of symmetry is the same vertical line. The graph crosses the y-axis at (0, c).
请记住：判别式 Δ = b² - 4ac 决定实数根的个数。y = ax² + bx + c 的顶点在 x = -b / (2a) 处，对称轴是同一竖直线。图像与 y 轴交于 (0, c)。
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