📚 Mastering Quadratic Equations | 精通二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, problem solving, and even in subjects such as physics and economics. This revision guide covers the main solution methods, the discriminant, graph sketching, and common exam mistakes.
二次方程是 IGCSE 数学中最重要的专题之一。它们出现在代数、图像、问题解决中，甚至也出现在物理和经济学等学科中。本复习指南涵盖主要解法、判别式、图像绘制以及常见的考试错误。
1. What is a Quadratic Equation? | 什么是二次方程？
rd form ax² + bx + c = 0, where a, b and c are constants, and a must not be equal to zero.
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、c 为常数，并且 a 不能等于零。
The highest power of the variable x is 2, which is why the word 'quadratic' is used. If a = 0, the equation becomes linear rather than quadratic.
变量 x 的最高次数是 2，这就是使用 "二次" 一词的原因。如果 a = 0，方程就变成了一次方程而不是二次方程。
Common examples include x² − 5x + 6 = 0, 2x² + 3x − 2 = 0 and x² − 9 = 0.
常见的例子包括 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 和 x² − 9 = 0。
2. Standard Form and Key Terms | 标准形式与关键术语
alled the quadratic term, bx is called the linear term, and c is called the constant term.
二次方程的标准形式为 ax² + bx + c = 0。其中 ax² 称为二次项，bx 称为一次项，c 称为常数项。
ro, but b or c can be zero. For example, x² − 9 = 0 has b = 0, and 2x² + 3x = 0 has c = 0.
系数 a 绝对不能为零，但 b 或 c 可以为零。例如，x² − 9 = 0 中 b = 0，而 2x² + 3x = 0 中 c = 0。
Always rearrange an equation into standard form before choosing a solution method. This makes it easier to identify a, b and c correctly.
在选择求解方法之前，务必先将方程整理为标准形式。这样可以更容易地正确确定 a、b 和 c。
3. Solving by Factorising | 因式分解法求解
Factorising is often the fastest method when the quadratic expression can be written as a product of two linear brackets.
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方法。
For a quadratic of the form x² + bx + c, look for two numbers that multiply to give c and add to give b.
对于 x² + bx + c 形式的二次式，寻找两个数，使它们的乘积等于 c，和等于 b。
− 2)(x − 3) = 0. Set each factor equal to zero: x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.
示例：解方程 x² − 5x + 6 = 0。因式分解得到 (x − 2)(x − 3) = 0。令每个因式等于零：x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。
When a is not equal to 1, you may need to split the middle term or test factor pairs systematically. For example, 2x² + 3x − 2 can be factorised as (2x − 1)(x + 2).
当 a 不等于 1 时，你可能需要拆分中项或系统地尝试因数对。例如，2x² + 3x − 2 可以因式分解为 (2x − 1)(x + 2)。
Always check your answer by expanding the brackets to confirm that the original quadratic expression is obtained.
始终通过展开括号来检验答案，确认得到原来的二次式。
4. Solving by Completing the Square | 配方法求解
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful when the quadratic does not factorise neatly and when you need the turning point of a graph.
配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。当二次式不能整齐地因式分解以及需要求图像顶点时，这种方法非常有用。
0. First write x² + 6x = −2. Then complete the square: (x + 3)² − 9 = −2, so (x + 3)² = 7.
示例：解方程 x² + 6x + 2 = 0。首先写成 x² + 6x = −2。然后配方：(x + 3)² − 9 = −2，所以 (x + 3)² = 7。
Taking square roots gives x + 3 = ±√7, so the solutions are x = −3 + √7 or x = −3 − √7.
两边开平方得到 x + 3 = ±√7，因此解为 x = −3 + √7 或 x = −3 − √7。
The number added inside the bracket is half the coefficient of x, and then its square is subtracted. For a ≠ 1, factor out a first, then complete the square inside the bracket.
括号内加上的数是 x 系数的一半，然后再减去它的平方。当 a ≠ 1 时，先提取系数 a，再在括号内完成配方。
5. The Quadratic Formula | 二次公式法
iable method that solves any quadratic equation written in standard form ax² + bx + c = 0.
二次公式是一种可靠的方法，可以求解任何写成标准形式 ax² + bx + c = 0 的二次方程。
x = (−b ± √b² − 4ac) ÷ 2a
Identify the values of a, b and c from the standard form, substitute them carefully, and simplify the result.
从标准形式中确定 a、b、c 的值，仔细代入，然后化简结果。
e a = 2, b = 3 and c = −2. The discriminant is b² − 4ac = 3² − 4 × 2 × (−2) = 9 + 16 = 25.
示例：解方程 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。判别式为 b² − 4ac = 3² − 4 × 2 × (−2) = 9 + 16 = 25。
Substitute into the formula: x = (−3 ± √25) ÷ 4 = (−3 ± 5) ÷ 4. This gives x = ½ or x = −2.
代入公式：x = (−3 ± √25) ÷ 4 = (−3 ± 5) ÷ 4。由此得到 x = ½ 或 x = −2。
The quadratic formula is especially useful when factorising is difficult or when the roots are irrational. Always simplify surds fully unless the question asks for decimal answers.
当因式分解困难或根为无理数时，二次公式尤其有用。除非题目要求小数值，否则务必完全化简根式。
6. Discriminant and Nature of Roots | 判别式与根的性质
he expression D = b² − 4ac. It is the part under the square root in the quadratic formula.
判别式是表达式 D = b² − 4ac，它是二次公式中平方根号下的部分。
tinct real roots. If D = 0, it has one repeated real root. If D < 0, it has no real roots.
如果 D > 0，方程有两个不相等的实根；如果 D = 0，方程有一个重实根；如果 D < 0，方程没有实根。
| Discriminant D = b² − 4ac | Nature of roots | 根的性质 |
|---|---|
| D > 0 | Two distinct real roots | 两个不相等的实根 |
| D = 0 | One repeated real root | 一个重实根 |
| D < 0 | No real roots | 没有实根 |
You can use the discriminant to determine the number of solutions without solving the equation fully. This is a common exam question.
你可以使用判别式在不完整求解方程的情况下判断解的个数。这是常见的考试题型。
7. Graphs of Quadratic Functions | 二次函数图像
c is a parabola. The sign of a determines whether the parabola opens upwards or downwards.
y = ax² + bx + c 的图像是一条抛物线。a 的符号决定抛物线是开口向上还是开口向下。
If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
如果 a > 0，抛物线开口向上，具有最小值点；
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