📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程全解析
where a ≠ 0. This topic is examined regularly because it links factorising, functions, graphs, and real-world applications.
二次方程贯穿 IGCSE 数学课程，从代数、图像到测量与应用题都经常出现。二次方程是指能化为 ax² + bx + c = 0（其中 a ≠ 0）的方程。这一专题经常出现在考试中，因为它把因式分解、函数、图像和实际应用紧密连接起来。
1. What is a Quadratic Equation? | 什么是二次方程？
b and c are constants, and a cannot be zero because otherwise the equation becomes linear.
二次方程是次数为 2 的多项式方程。变量 x 的最高次数是 2，方程总可以写成标准形式 ax² + bx + c = 0。a、b、c 是常数，且 a 不能为零，否则方程就变成了一次方程。
ax² + bx + c = 0, where a ≠ 0
ratic equations. The equation x³ + x² = 0 is not quadratic because the highest power is 3.
例如，x² − 5x + 6 = 0 和 3x² + 2x − 1 = 0 都是二次方程。x³ + x² = 0 不是二次方程，因为最高次数是 3。
2. Standard Form and Key Vocabulary | 标准形式与关键术语
0. The expression on the left has three terms: the quadratic term ax², the linear term bx, and the constant term c. The coefficients a, b and c can be positive or negative.
在 IGCSE 考题中，你通常会看到二次方程写成 ax² + bx + c = 0。左边的表达式有三项：二次项 ax²、一次项 bx 和常数项 c。系数 a、b、c 可以是正数，也可以是负数。
- Leading coefficient: the coefficient of x², which is a. | 首项系数：x² 的系数，即 a。
- Linear coefficient: the coefficient of x, which is b. | 一次项系数：x 的系数，即 b。
- Constant term: the number c with no variable. | 常数项：不含变量的数 c。
- Roots or solutions: the values of x that make the equation true. | 根或解：使方程成立的 x 的值。
Many students confuse the coefficient b with the term bx. Always rewrite the equation in standard form before identifying a, b and c.
许多学生容易把系数 b 和一次项 bx 混淆。在确定 a、b、c 之前，一定要先把方程写成标准形式。
3. Solving by Factorising | 因式分解法
Factorising is the fastest method when the quadratic expression can be factorised easily. The key idea is that if the product of two expressions is zero, then at least one of the expressions must be zero. This is called the zero product property.
当二次式容易因式分解时，因式分解法是最快的方法。核心思想是：如果两个式子的乘积等于零，那么其中至少有一个式子必须等于零。这称为零乘积性质。
If (x − p)(x − q) = 0, then x = p or x = q.
3). Then set each factor equal to zero: x − 2 = 0 gives x = 2, and x − 3 = 0 gives x = 3.
解 x² − 5x + 6 = 0，先将左边因式分解为 (x − 2)(x − 3)。然后令每个因式等于零：x − 2 = 0 得 x = 2，x − 3 = 0 得 x = 3。
- Write the equation in standard form. | 将方程写成标准形式。
- Factorise the quadratic expression. | 对二次式进行因式分解。
- Set each bracket equal to zero and solve. | 令每个括号等于零并求解。
nal equation. For example, when x = 2, the equation becomes 4 − 10 + 6 = 0, which is true.
一定要把答案代回原方程检验。例如当 x = 2 时，方程变为 4 − 10 + 6 = 0，成立。
4. Solving by the Quadratic Formula | 公式法
+ c = 0. You should memorise the formula and know how to substitute a, b and c correctly.
公式法适用于所有二次方程，即使二次式不能因式分解也能使用。它是由一般二次方程 ax² + bx + c = 0 配方推导出来的。你应当记住这个公式，并且能够正确代入 a、b、c。
x = (−b ± √b² − 4ac) ÷ 2a
3, c = −2. Substitute into the formula: x = (−3 ± √(3² − 4 × 2 × (−2))) ÷ (2 × 2) = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4.
例如：解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。代入公式：x = (−3 ± √(3² − 4 × 2 × (−2))) ÷ (2 × 2) = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4。
3 − 5) ÷ 4 = −2. In IGCSE exams, show each substitution step clearly to gain method marks.
由此得到两个解：x = (−3 + 5) ÷ 4 = 1/2 和 x = (−3 − 5) ÷ 4 = −2。在 IGCSE 考试中，要清楚地写出每一步代入过程，以获得方法分。
5. Completing the Square | 配方法
Completing the square is useful when a quadratic cannot be factorised easily and when you need to find the vertex of a parabola. The goal is to rewrite ax² + bx + c in the form a(x + p)² + q.
当二次式不容易因式分解，或者需要求抛物线的顶点时，配方法非常有用。其目标是把 ax² + bx + c 改写成 a(x + p)² + q 的形式。
x² + bx = (x + b/2)² − (b/2)²
, which is 3, and square it to get 9. Write x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8.
例如：用配方法解 x² + 6x + 1 = 0。取 6 的一半为 3，平方得 9。于是 x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8。
Set (x + 3)² − 8 = 0, so (x + 3)² = 8. Taking square roots gives x + 3 = ±√8 = ±2√2, so x = −3 ± 2√2.
令 (x + 3)² − 8 = 0，则 (x + 3)² = 8。两边开方得 x + 3 = ±√8 = ±2√2，所以 x = −3 ± 2√2。
t the vertex of y = x² + 6x + 1 is at (−3, −8), which is very helpful for graph questions.
配方法还能告诉你 y = x² + 6x + 1 的顶点是 (−3, −8)，这对图像题非常有帮助。
6. The Discriminant and Number of Roots | 判别式与根的个数
The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells you how many real roots a quadratic equation has without solving the equation completely.
二次公式中的表达式 b² − 4ac 称为判别式。它可以在不完整解方程的情况下，告诉你二次方程有多少个实数根。
Δ = b² − 4ac
| Discriminant value | 判别式的值 | Nature of roots | 根的性质 |
|---|---|
| Δ > 0 | Two distinct real roots | 两个不相等的实数根 |
| Δ = 0 | One repeated real root | 一个重根 |
| Δ < 0 | No real roots | 无实数根 |
16 = 25. Since 25 > 0, there are two distinct real roots, which we found to be 1/2 and −2.
例如，在 2x² + 3x − 2 = 0 中，判别式为 3² − 4 × 2 × (−2) = 9 + 16 = 25。因为 25 > 0，所以有两个不相等的实数根，也就是我们求出的 1/2 和 −2。
7. Graphical Interpretation | 图像解释
where the parabola crosses the x-axis. These points are called the x-intercepts or roots.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解就是抛物线与 x 轴交点的横坐标。这些点称为 x 轴截距或根。
If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The line of symmetry is vertical and passes through the vertex.
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。对称轴是一条经过顶点的竖直线。
x = −b ÷ 2a
= 0 means it touches the x-axis once, and Δ < 0 means it does not cross the x-axis at all.
判别式也与图像有关：Δ > 0 表示抛物线与 x 轴有两个交点，Δ = 0 表示抛物线与 x 轴相切一次，Δ < 0 表示抛物线与 x 轴没有交点。
8. Word Problems and Applications | 应用题
IGCSE exam questions often hide a quadratic equation inside a real-life problem. You need to define a variable, form an equation, solve it, and then interpret the answer in context.
IGCSE 考试题常常把二次方程隐藏在现实问题中。你需要设出变量、建立方程、求解，然后根据题意解释答案。
². Let the width be x cm. Then the length is x + 3 cm. The area equation is x(x + 3) = 40.
例如：一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x cm，则长为 x + 3 cm。面积方程为 x(x + 3) = 40。
nce a width cannot be negative, x = 5. Therefore the width is 5 cm and the length is 8 cm.
展开得 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0。可能的 x 值是 −8 和 5。因为宽度不能为负，所以 x = 5。因此宽为 5 cm，长为 8 cm。
Always reject solutions that are impossible in the given situation, such as negative lengths, times, or areas.
一定要舍去题目情境中不可能的解，例如负的长度、时间或面积。
9. Common Mistakes | 常见错误
Many students lose marks on quadratic equations through careless errors rather than a lack of understanding. Knowing the common mistakes will help you avoid them in the exam.
许多学生在二次方程上丢分，不是因为不理解，而是因为粗心。了解常见错误可以帮助你在考试中避免这些问题。
- Forgetting to set the equation equal to 0 before factorising. | 因式分解前忘记先把方程化为等于 0。
- Dividing both sides by x and losing the root x = 0. | 两边同时除以 x，从而丢失 x = 0 这个根。
- Making sign errors when substituting negative values into the formula. | 将负值代入公式时出现符号错误。
- Misidentifying a, b and c when the equation is not in standard form. | 当方程不是标准形式时，错误识别 a、b、c。
- Forgetting to check whether the roots satisfy the original equation. | 忘记检验根是否满足原方程。
x and get x = 4. The correct method is x² − 4x = 0, then x(x − 4) = 0, so x = 0 or x = 4.
例如，对于方程 x² = 4x，常见错误是两边除以 x，得到 x = 4。正确的方法是把 x² − 4x = 0 写成 x(x − 4) = 0，所以 x = 0 或 x = 4。
10. Exam Style Tips | 考试技巧
In IGCSE exams, method marks are often awarded for correct substitution and factorising, even if the final answer is wrong. Always show your working step by step, especially when using the quadratic formula.
在 IGCSE 考试中，即使最终答案错误，正确的代入和因式分解步骤通常也能获得方法分。一定要一步一步写出解题过程，尤其是在使用公式法时。
- Write the standard form first and identify a, b and c clearly. | 先写出标准形式，并清楚地标出 a、b、c。
- Use brackets when substituting negative numbers into the formula. | 把负数代入公式时使用括号。
- Check your roots by substituting them into the original equation or by using the discriminant. | 通过代回原方程或利用判别式来检验根。
- If the question asks for exact answers, leave your answer in surd form such as −3 ± 2√2, not a decimal. | 如果题目要求精确答案，请把答案保留为根式形式，如 −3 ± 2√2，而不是写成小数。
- In word problems, re-read the final sentence to ensure your answer makes sense in context. | 在应用题中，重新读一遍题目的最后一句话，确保答案符合题意。
Practice a mixture of factorising, formula, completing the square, and graphical questions so that you can choose the most efficient method under timed conditions.
混合练习因式分解、公式法、配方法和图像题，这样你才能在限时条件下选择最高效的方法。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图