📚 Solving Quadratic Equations | 解二次方程
A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, geometry, and many real-life problems. Understanding how to solve quadratic equations confidently will help you score high in your Edexcel exam.
二次方程是 IGCSE 数学中最重要的内容之一。它出现在代数、图像、几何以及许多实际问题中。自信地掌握解二次方程的方法，将帮助你在 Edexcel 考试中取得高分。
1. What Is a Quadratic Equation? | 什么是二次方程？
a, b and c are numbers, and a ≠ 0. The highest power of x is 2, which is why it is called "quadratic".
二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 是数字，且 a ≠ 0。未知数 x 的最高次数为 2，因此被称为“二次”。
For example: 2x² + 3x - 5 = 0, x² - 4 = 0, and 3x² = 6x are all quadratic equations.
例如：2x² + 3x - 5 = 0、x² - 4 = 0 和 3x² = 6x 都是二次方程。
Some equations can be rearranged into this standard form before solving. Always make sure one side is zero before using factorisation or the quadratic formula.
有些方程需要先变形为标准形式再求解。在使用因式分解或求根公式前，一定要确保一边为零。
2. Solving by Factorisation | 因式分解法
two linear factors. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0.
当二次表达式可以写成两个一次因式的乘积时，因式分解是最快捷的方法。例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0。
then at least one factor must be zero. So x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.
如果两个因式的乘积为零，那么至少有一个因式为零。所以 x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。
Steps for factorising a quadratic ax² + bx + c:
因式分解二次式 ax² + bx + c 的步骤：
- Find two numbers that multiply to give ac and add to give b.
- 找到两个数，它们的乘积等于 ac，和等于 b。
- Split the middle term using these two numbers.
- 用这两个数拆分中间项。
- Factor by grouping.
- 分组因式分解。
- Set each bracket equal to zero.
- 令每个括号等于零。
3 = 0, so x(2x + 1) + 3(2x + 1) = 0, giving (2x + 1)(x + 3) = 0, hence x = -1/2 or x = -3.
例如：2x² + 7x + 3 = 0。两个数：1 和 6（1 × 6 = 6 = ac，1 + 6 = 7 = b）。于是 2x² + 1x + 6x + 3 = 0，即 x(2x + 1) + 3(2x + 1) = 0，得到 (2x + 1)(x + 3) = 0，因此 x = -1/2 或 x = -3。
If the coefficient of x² is 1, you can simply look for two numbers that multiply to c and add to b.
如果 x² 的系数为 1，你只需找两个数，它们相乘等于 c，相加等于 b。
3. The Quadratic Formula | 求根公式
When a quadratic cannot be factorised easily, the quadratic formula gives the solution directly.
当二次方程不易因式分解时，求根公式可以直接给出解。
x = (−b ± √b² − 4ac) / (2a)
x = (−b ± √b² − 4ac) / (2a)
e from the equation ax² + bx + c = 0. Substitute the values into the formula and simplify.
其中 a、b、c 来自方程 ax² + bx + c = 0。将数值代入公式并化简。
For example, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2.
例如，解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。
Substitute: x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √9 + 16) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4.
代入：x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √9 + 16) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4。
So x = (−3 + 5)/4 = 1/2, or x = (−3 − 5)/4 = −2. The solutions are x = 1/2 and x = −2.
所以 x = (−3 + 5)/4 = 1/2，或 x = (−3 − 5)/4 = −2。解为 x = 1/2 和 x = −2。
Remember to check if b² − 4ac is negative before using the formula. If it is negative, there are no real solutions.
在使用公式前，记得检查 b² − 4ac 是否为负。如果为负，则没有实数解。
4. Completing the Square | 配方法
. This is useful for solving equations and finding the turning point of a quadratic graph.
配方法将二次式改写成 p(x + q)² + r = 0 的形式。这在解方程和求二次图像顶点时非常有用。
e by a first. Then take half of the coefficient of x, square it, and add and subtract it inside the equation.
从 ax² + bx + c = 0 开始。如果 a ≠ 1，先除以 a。然后取 x 系数的一半，平方，并在等式中加上和减去这个平方。
Example: x² + 6x + 2 = 0. Half of 6 is 3, square is 9. So x² + 6x + 9 − 9 + 2 = 0 ⇒ (x + 3)² − 7 = 0. Then (x + 3)² = 7, so x + 3 = ±√7, hence x = −3 ± √7.
例如：x² + 6x + 2 = 0。6 的一半是 3，平方为 9。所以 x² + 6x + 9 − 9 + 2 = 0 ⇒ (x + 3)² − 7 = 0。于是 (x + 3)² = 7，所以 x + 3 = ±√7，因此 x = −3 ± √7。
s 2(x² + 4x) + 5 = 0. Then complete the square inside the bracket: 2[(x + 2)² − 4] + 5 = 0 ⇒ 2(x + 2)² − 8 + 5 = 0 ⇒ 2(x + 2)² − 3 = 0. So (x + 2)² = 3/2, giving x = −2 ± √3/2.
当 x² 的系数不是 1 时，先提取出来。例如，2x² + 8x + 5 = 0 变成 2(x² + 4x) + 5 = 0。然后在括号内配方：2[(x + 2)² − 4] + 5 = 0 ⇒ 2(x + 2)² − 8 + 5 = 0 ⇒ 2(x + 2)² − 3 = 0。所以 (x + 2)² = 3/2，得到 x = −2 ± √3/2。
Completing the square also helps you find the minimum or maximum point of a quadratic curve.
配方法还能帮助你找到二次曲线的最小值或最大值点。
5. The Discriminant | 判别式
In the quadratic formula, the expression inside the square root, b² − 4ac, is called the discriminant. It tells you how many real solutions the equation has.
在求根公式中，根号内的表达式 b² − 4ac 被称为判别式。它告诉我们方程有多少个实数解。
- If b² − 4ac > 0, there are two distinct real solutions.
- 如果 b² − 4ac > 0，方程有两个不同的实数解。
- If b² − 4ac = 0, there is exactly one real solution (a repeated root).
- 如果 b² − 4ac = 0，方程恰好有一个实数解（重根）。
- If b² − 4ac < 0, there are no real solutions.
- 如果 b² − 4ac < 0，方程没有实数解。
4 = 0 has discriminant (−4)² − 4 × 1 × 4 = 16 − 16 = 0, so it has one repeated root x = 2.
例如，x² − 4x + 4 = 0 的判别式为 (−4)² − 4 × 1 × 4 = 16 − 16 = 0，所以它有一个重根 x = 2。
The discriminant also tells you how many x-intercepts a quadratic graph has. A positive discriminant means two intercepts with the x-axis, zero means one intercept, and negative means no intercept.
判别式还告诉我们二次图像与 x 轴的交点个数。判别式为正表示有两个交点，为零表示有一个交点，为负表示没有交点。
6. Solving by Drawing Graphs | 图像法
ph of y = ax² + bx + c. The solutions are the x-values where the graph crosses the x-axis.
有时你可能会被要求通过读取 y = ax² + bx + c 的图像来解二次方程。解就是图像与 x 轴交点的 x 值。
To draw a quadratic graph, plot at least five points, including the turning point and points on both sides of it. Then draw a smooth curve through the points.
要画二次图像，至少绘制五个点，包括顶点和两侧的点。然后用平滑曲线穿过这些点。
2x − 3 crosses the x-axis at x = −1 and x = 3. Those are the solutions to x² − 2x − 3 = 0.
例如，y = x² − 2x − 3 的图像在 x = −1 和 x = 3 处穿过 x 轴。这就是方程 x² − 2x − 3 = 0 的解。
m y = ax² + bx + c, you may need to rearrange it or solve a pair of equations graphically.
如果方程不是 y = ax² + bx + c 的形式，你可能需要重新整理，或通过图像解一对方程组。
7. Quadratic Graphs and the Turning Point | 二次图像与顶点
The graph of a quadratic function is a parabola. It is symmetric about a vertical line, and the highest or lowest point is called the turning point.
二次函数的图像是抛物线。它关于一条竖直线对称，最高点或最低点称为顶点。
If the coefficient of x² is positive, the parabola opens upwards and the turning point is a minimum. If the coefficient is negative, it opens downwards and the turning point is a maximum.
如果 x² 的系数为正，抛物线开口向上，顶点是最小值。如果系数为负，抛物线开口向下，顶点是最大值。
int is x = −b / (2a). To find the y-coordinate, substitute this x-value into the equation.
顶点的 x 坐标为 x = −b / (2a)。要得到 y 坐标，将此 x 值代入原方程。
= −4, so x = 4 / (2 × 1) = 2. Then y = 2² − 4 × 2 + 5 = 1. So the turning point is (2, 1).
例如，y = x² − 4x + 5。这里 a = 1，b = −4，所以 x = 4 / (2 × 1) = 2。然后 y = 2² − 4 × 2 + 5 = 1。因此顶点为 (2, 1)。
ng completing the square, y = (x − 2)² + 1, which directly shows the turning point (2, 1).
利用配方法，y = (x − 2)² + 1，直接显示出顶点 (2, 1)。
8. Word Problems with Quadratics | 二次方程应用题
Many real-world problems produce quadratic equations. You must first translate the problem into an equation, then solve it, and finally check that your answers make sense.
许多现实问题会产生二次方程。你必须先将问题转化为方程，然后求解，最后检查答案是否合理。
th × width = area; for motion, height = initial height + velocity × time − 5t² (on Earth).
常见类型包括面积问题、抛体运动和数字谜题。对面积，长 × 宽 = 面积；对运动，高度 = 初始高度 + 速度 × 时间 − 5t²（在地球上）。
e than its width. Its area is 24 cm². Let width = x, then length = x + 2. So x(x + 2) = 24 ⇒ x² + 2x − 24 = 0 ⇒ (x + 6)(x − 4) = 0 ⇒ x = 4 or x = −6. Since width cannot be negative, width = 4 cm and length = 6 cm.
例：一个长方形的长比宽多 2 cm，面积为 24 cm²。设宽为 x，则长为 x + 2。所以 x(x + 2) = 24 ⇒ x² + 2x − 24 = 0 ⇒ (x + 6)(x − 4) = 0 ⇒ x = 4 或 x = −6。因为宽不能为负，所以宽 = 4 cm，长 = 6 cm。
Always reject negative or impossible answers in word problems, even if they are mathematically valid.
在应用题中，即使负根或不合理的根在数学上成立，也要舍弃。
9. Common Mistakes to Avoid | 常见错误提醒
Students often lose marks in quadratic questions because of small errors. Here are some common mistakes and how to avoid them.
学生在二次方程题目中常因小错误而失分。以下是一些常见错误及避免方法。
- Forgetting to write the equation in the form ax² + bx + c = 0 before factorising.
- 在因式分解前忘记把方程写成 ax² + bx + c = 0 的形式。
- Using the quadratic formula incorrectly because the signs of b or c are wrong.
- 因为 b 或 c 的符号错误而错用求根公式。
- Dropping the ± sign when taking square roots.
- 在开平方时忽略 ± 号。
- Thinking that (a + b)² = a² + b². This is false.
- 错误认为 (a + b)² = a² + b²。这是错误的。
- Not checking solutions by substituting back into the original equation.
- 不将解代回原方程进行检验。
To avoid these mistakes, write carefully, show every step, and practice regularly.
为了避免这些错误，请书写工整，展示每一步，并经常练习。
10. Exam Tips for Edexcel | Edexcel 考试建议
In the Edexcel IGCSE exam, quadratic questions appear in both Paper 1 and Paper 2, and may be worth 2 to 6 marks. Method marks are often awarded even if your final answer is wrong.
在 Edexcel IGCSE 考试中，二次方程问题出现在 Paper 1 和 Paper 2 中，可能占 2 到 6 分。即使最终答案错误，常常也会给方法分。
Always show clear working. If you use a calculator, write down the substituted expression before giving the answer.
务必展示清晰的计算过程。如果使用计算器，先写下代入后的表达式再给出答案。
Know when to use each method:
知道何时使用哪种方法：
| Method | When to use |
| Factorisation | 因式分解 | When the quadratic is easy to factorise, coefficient of x² is often 1 or a small number. |
| Quadratic formula | 求根公式 | When factorisation is difficult or impossible. |
| Completing the square | 配方法 | When you need the turning point or when a question asks for a solution in surd form. |
| Graph | 图像 | When the question gives a graph or asks you to draw one. |
A calculator can check your roots quickly, but you still need to show the algebraic method for full marks.
计算器可以快速检验你的根，但你仍需展示代数方法才能得满分。
11. Practice Problems | 练习题目
Try these problems on your own after reading this article:
阅读本文后，请独立尝试这些题目：
- Solve x² − 7x + 12 = 0 by factorisation.
- 用因式分解解 x² − 7x + 12 = 0。
- Solve 3x² + 2x − 1 = 0 using the quadratic formula.
- 用求根公式解 3x² + 2x − 1 = 0。
- Write x² − 8x + 6 in the form (x − p)² + q, and find the minimum point.
- 将 x² − 8x + 6 写成 (x − p)² + q 的形式，并求最小值点。
- Find the discriminant of x² + 5x + 3 = 0, and state how many real roots it has.
- 求 x² + 5x + 3 = 0 的判别式，并说明它有几个实数根。
- The sum of a number and its square is 42. Find the number.
- 一个数与它的平方之和为 42，求这个数。
Answers are easy to check with a calculator. If you get stuck, review the relevant section above.
答案很容易用计算器检验。如果卡住，请回顾上面相关章节。
12. Conclusion | 总结
Solving quadratic equations is a skill that connects many parts of IGCSE Mathematics. Master the four methods: factorisation, quadratic formula, completing the square, and graphing. Understand the discriminant, and always show your working in the exam.
解二次方程是一项连接 IGCSE 数学许多章节的技能。掌握四种方法：因式分解、求根公式、配方法和图像法。理解判别式，并在考试中始终展示你的解题过程。
With regular practice, you will become fast and accurate. Keep a list of mistakes you have made and review it before the test.
通过定期练习，你将变得又快又准确。把你犯过的错误列一个清单，并在考试前复习。
Good luck with your Edexcel IGCSE Mathematics exam!
祝你在 Edexcel IGCSE 数学考试中好运！
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