📚 Solving Quadratic Equations | 解一元二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, geometry, and even real-life problem solving. This guide will take you through every method you need to solve a quadratic equation confidently, step by step.
一元二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、图像、几何甚至实际生活问题中。本指南将带你逐步掌握解决一元二次方程所需的每一种方法，让你自信应对考试。
1. What Is a Quadratic Equation? | 什么是一元二次方程
A quadratic equation is a polynomial equation of degree 2. This means the highest power of the variable is 2. The general form is written as:
一元二次方程是次数为 2 的多项式方程，这意味着变量的最高次数是 2。它的一般形式写作：
ax² + bx + c = 0
Here, a, b, and c are constants, and a ≠ 0. If a were zero, the equation would become linear, not quadratic.
其中 a、b、c 是常数，且 a ≠ 0。如果 a 等于零，方程就会变成一次方程，而不是二次方程。
For example, x² - 5x + 6 = 0 is quadratic, while 2x + 3 = 0 is not.
例如，x² - 5x + 6 = 0 是二次方程，而 2x + 3 = 0 不是。
2. Standard Form and Key Features | 标准形式与关键特征
0. This makes it easier to factorise, complete the square, or apply the quadratic formula.
在解任何一元二次方程之前，你应该先将其整理成标准形式 ax² + bx + c = 0。这样更容易进行因式分解、配方或使用求根公式。
Key features of a quadratic expression ax² + bx + c include:
二次表达式 ax² + bx + c 的关键特征包括：
- The coefficient of x² (a) determines whether the graph opens upward (a > 0) or downward (a < 0).
- x² 的系数 (a) 决定图像开口向上 (a > 0) 还是向下 (a < 0)。
- The constant term c gives the y-intercept of the graph.
- 常数项 c 给出图像与 y 轴的交点。
- The roots are the x-values where the graph crosses the x-axis.
- 根 是图像与 x 轴交点对应的 x 值。
Always check that the equation is in the correct order: x² term first, then x term, then constant.
始终检查方程是否按正确顺序排列：先 x² 项，再 x 项，最后常数项。
3. Solving by Factorisation | 因式分解法
Factorisation is often the quickest method when the quadratic has simple integer factors. The idea is to write ax² + bx + c as a product of two brackets.
当二次方程具有简单的整数因子时，因式分解通常是最快的方法。其核心思想是将 ax² + bx + c 写成两个括号的乘积。
0. We look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3.
考虑方程 x² - 5x + 6 = 0。我们需要找到两个数，它们相乘等于 6，相加等于 -5。这两个数是 -2 和 -3。
(x - 2)(x - 3) = 0
of two factors is zero, then at least one factor must be zero. So x - 2 = 0 or x - 3 = 0.
如果两个因子的乘积为零，那么至少有一个因子为零。所以 x - 2 = 0 或 x - 3 = 0。
Therefore, the solutions are x = 2 and x = 3.
因此，解为 x = 2 和 x = 3。
numbers that multiply to 6 and add to 7. These are 1 and 6. Then rewrite the middle term:
对于首项系数不为 1 的二次方程，例如 2x² + 7x + 3 = 0，因式分解需要更加小心。将 a 和 c 相乘 (2 × 3 = 6)，找到两个数相乘等于 6 且相加等于 7，它们是 1 和 6。然后重写中间项：
2x² + 1x + 6x + 3 = 0
factorise: x(2x + 1) + 3(2x + 1) = 0, giving (2x + 1)(x + 3) = 0. Hence x = -½ or x = -3.
分组并提取公因式：x(2x + 1) + 3(2x + 1) = 0，得到 (2x + 1)(x + 3) = 0。因此 x = -½ 或 x = -3。
4. Solving by Completing the Square | 配方法
s is useful for solving equations that do not factorise easily, and it also helps in sketching graphs.
配方法将二次方程转化为 (x + p)² = q 的形式。这在解不易因式分解的方程时非常有用，也有助于绘制图像。
by halving the coefficient of x, which is 6, to get 3. Write (x + 3)² and subtract 3² = 9:
以方程 x² + 6x + 2 = 0 为例。首先将 x 的系数 6 除以 2，得到 3。写成 (x + 3)²，然后减去 3² = 9：
(x + 3)² - 9 + 2 = 0
Simplify: (x + 3)² - 7 = 0, so (x + 3)² = 7.
化简： (x + 3)² - 7 = 0，所以 (x + 3)² = 7。
Take the square root of both sides: x + 3 = ±√7. Therefore x = -3 ± √7.
两边开平方：x + 3 = ±√7。因此 x = -3 ± √7。
ple, for 2x² - 8x + 5 = 0, divide by 2 to get x² - 4x + 2.5 = 0, then complete the square.
对于 a > 1 的二次方程，先两边除以 a。例如，对于 2x² - 8x + 5 = 0，除以 2 得到 x² - 4x + 2.5 = 0，然后再配方。
5. The Quadratic Formula | 求根公式
ethod that works for any quadratic equation. For ax² + bx + c = 0, the roots are given by:
求根公式是一种适用于任何一元二次方程的通用方法。对于 ax² + bx + c = 0，根由下式给出：
x = (-b ± √b² - 4ac) / 2a
Let's solve 2x² - 3x - 2 = 0 using the formula. Here a = 2, b = -3, c = -2.
让我们用公式解 2x² - 3x - 2 = 0。其中 a = 2，b = -3，c = -2。
First calculate the discriminant b² - 4ac = (-3)² - 4(2)(-2) = 9 + 16 = 25.
首先计算判别式 b² - 4ac = (-3)² - 4(2)(-2) = 9 + 16 = 25。
Then substitute: x = (3 ± √25) / 4 = (3 ± 5) / 4.
然后代入：x = (3 ± √25) / 4 = (3 ± 5) / 4。
(3 + 5)/4 = 2, or x = (3 - 5)/4 = -½. This matches the factorisation result from earlier.
所以 x = (3 + 5)/4 = 2，或 x = (3 - 5)/4 = -½。这与之前的因式分解结果一致。
The formula is especially helpful when factorisation is difficult or impossible, such as when the roots are irrational or complex.
当因式分解困难或无法进行时，例如根为无理数或复数时，这个公式特别有帮助。
6. The Discriminant | 判别式
The expression b² - 4ac inside the quadratic formula is called the discriminant. It tells us how many real roots the quadratic equation has, without solving it fully.
求根公式中的 b² - 4ac 称为判别式。它无需完全解方程，就能告诉我们一元二次方程有多少个实数根。
There are three cases:
有三种情况：
| Discriminant Δ = b² - 4ac | Number of Real Roots | Graph Intersection with x-axis |
| Δ > 0 | Two distinct real roots | Two points |
| Δ = 0 | One repeated real root | One point (touches) |
| Δ < 0 | No real roots | No intersection |
判别式 Δ = b² - 4ac
判别式 Δ = b² - 4ac 的三种情况：Δ > 0 表示有两个不同的实数根；Δ = 0 表示有一个重根；Δ < 0 表示没有实数根。
For example, the equation x² + 2x + 5 = 0 has Δ = 4 - 20 = -16, so it has no real roots.
例如，方程 x² + 2x + 5 = 0 的判别式 Δ = 4 - 20 = -16，因此它没有实数根。
Knowing the discriminant also helps you decide which method to use. If Δ is a perfect square, the quadratic factorises neatly.
了解判别式还有助于你决定使用哪种方法。如果 Δ 是完全平方数，则二次方程可以简洁地因式分解。
7. Solving Word Problems | 应用题求解
Many real-world problems can be modelled by quadratic equations. Common examples include area problems, projectile motion, and number puzzles.
许多现实问题都可以用二次方程建模。常见的例子包括面积问题、抛体运动和数字谜题。
Consider a rectangle whose length is 3 cm longer than its width. If the area is 40 cm², find the width.
考虑一个长方形，其长比宽长 3 cm。如果面积为 40 cm²，求宽。
Let the width be x cm. Then the length is (x + 3) cm. The equation is:
设宽为 x cm，则长为 (x + 3) cm。方程为：
x(x + 3) = 40
Expand: x² + 3x - 40 = 0. Factorise: (x + 8)(x - 5) = 0.
展开：x² + 3x - 40 = 0。因式分解：(x + 8)(x - 5) = 0。
So x = -8 or x = 5. Since a width cannot be negative, the width is 5 cm.
所以 x = -8 或 x = 5。由于宽度不能为负数，所以宽为 5 cm。
Always check whether the solutions make sense in the context of the problem. Negative lengths, times, or counts are usually rejected.
始终检查解在问题情境中是否有意义。负数长度、时间或数量通常会被舍去。
8. Sketching Quadratic Graphs | 二次函数图像
x² + bx + c is a parabola. Understanding its shape and key points helps you solve equations graphically.
二次函数 y = ax² + bx + c 的图像是一条抛物线。理解其形状和关键点有助于用图像法解方程。
The important features are:
重要特征包括：
- Shape: If a > 0, the parabola opens upward (U-shaped). If a < 0, it opens downward (n-shaped).
- 形状：如果 a > 0，抛物线开口向上（U 形）。如果 a < 0，开口向下（n 形）。
- Axis of symmetry: x = -b / 2a.
- 对称轴：x = -b / 2a。
- Vertex: The turning point, located on the axis of symmetry.
- 顶点：转向点，位于对称轴上。
- y-intercept: (0, c).
- y 轴截距：(0, c)。
- Roots: Solutions to ax² + bx + c = 0, where the graph crosses the x-axis.
- 根：ax² + bx + c = 0 的解，即图像与 x 轴的交点。
ertex, complete the square or use the formula x = -b / 2a, then substitute back to find y.
要找到顶点，可以使用配方法或公式 x = -b / 2a，然后代回求 y。
Sketching the graph is a good way to check your algebraic solutions for reasonableness.
绘制图像是检查代数解是否合理的好方法。
9. Common Mistakes | 常见错误
Students often lose marks on quadratic equations due to avoidable mistakes. Here are the most common ones and how to avoid them.
学生通常因为可避免的错误而在一元二次方程上失分。以下是最常见的错误以及如何避免它们。
- Forgetting to rearrange into standard form: Always ensure the equation equals 0 before factorising or using the formula.
- 忘记整理成标准形式：在因式分解或使用公式之前，始终确保方程等于 0。
- Incorrect signs: When a, b, or c are negative, substitute carefully into the formula.
- 符号错误：当 a、b 或 c 为负数时，代入公式时要格外小心。
- Missing the ± sign: The square root operation gives two answers unless the discriminant is zero.
- 遗漏 ± 符号：除非判别式为零，否则开平方会给出两个答案。
- Dividing by a variable: Never divide both sides by x, as this loses the root x = 0.
- 除以变量：永远不要两边同时除以 x，因为这会丢失根 x = 0。
- Ignoring negative roots in word problems: Reject them only if they make no sense in context.
- 在应用题中忽略负根：只有当负根在情境中无意义时才舍弃。
A reliable way to check your work is to substitute your answers back into the original equation.
检查工作的一种可靠方法是将你的答案代回原方程。
10. Practice Questions | 练习题
Now it's time to apply what you have learned. Try these questions without looking at the solutions, then check your answers.
现在是时候应用你所学的内容了。尝试不看解答做这些题，然后检查你的答案。
- Solve x² - 7x + 12 = 0 by factorisation.
- 用因式分解法解 x² - 7x + 12 = 0。
- Solve 3x² + 5x - 2 = 0 using the quadratic formula.
- 用求根公式解 3x² + 5x - 2 = 0。
- Solve x² + 2x - 4 = 0 by completing the square. Leave your answer in surd form.
- 用配方法解 x² + 2x - 4 = 0。答案保留根号形式。
- Find the discriminant of 2x² + 4x + 1 = 0 and state the number of real roots.
- 求 2x² + 4x + 1 = 0 的判别式，并说明实数根的个数。
- The product of two consecutive positive integers is 56. Find the integers.
- 两个连续正整数的乘积是 56。求这两个整数。
Answers: 1) x = 3, x = 4. 2) x = ½, x = -2. 3) x = -1 ± √5. 4) Δ = 8, two real roots. 5) 7 and 8.
答案：1) x = 3，x = 4。2) x = ½，x = -2。3) x = -1 ± √5。4) Δ = 8，两个实数根。5) 7 和 8。
If you make a mistake, review the relevant section and try again. Consistent practice is the key to mastering quadratic equations.
如果你犯了错误，请复习相关章节并再次尝试。坚持练习是掌握一元二次方程的关键。
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