📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in many different contexts, from pure algebra to real-world problem solving. In this revision guide, we will explore all the methods you need to solve them confidently.
二次方程是 IGCSE 数学中最重要的主题之一。它们出现在许多不同的情境中，从纯代数到现实世界的问题解决。在本复习指南中，我们将探索所有你需要自信解决二次方程的方法。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree 2. This means the highest power of the variable (usually x) is 2. The general form is:
二次方程是次数为 2 的多项式方程。这意味着变量（通常是 x）的最高次幂是 2。其一般形式为：
ax² + bx + c = 0, (a ≠ 0)
Here, a, b, and c are constants, with a being the coefficient of x², b the coefficient of x, and c the constant term. Because a ≠ 0, the equation is truly quadratic; if a = 0, it becomes linear.
这里，a、b 和 c 是常数，其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。因为 a ≠ 0，方程才真正是二次的；如果 a = 0，它就变成一次方程。
For example, 3x² - 5x + 2 = 0 is a quadratic equation, while 2x + 7 = 0 is not.
例如，3x² - 5x + 2 = 0 是二次方程，而 2x + 7 = 0 不是。
2. Standard Form | 标准形式
To solve a quadratic equation, it is often convenient to rewrite it in standard form: ax² + bx + c = 0. This means all terms are on one side, and the right-hand side is zero.
为了解二次方程，通常方便将其重写为标准形式：ax² + bx + c = 0。这意味着所有项都在一边，右边为零。
For example, the equation 2x² = 3x - 4 can be rearranged as:
例如，方程 2x² = 3x - 4 可以重新排列为：
2x² - 3x + 4 = 0
Always check that the equation is in this form before applying any solving technique. If there are fractions or brackets, expand and simplify first.
在应用任何求解技巧之前，务必检查方程是否为此形式。如果有分数或括号，先展开并化简。
- Expand brackets: Remove parentheses by multiplying out.
- Combine like terms: Add or subtract terms with the same power of x.
- Move all terms to one side: Ensure the right side equals zero.
- 展开括号：通过乘法去除括号。
- 合并同类项：将具有相同 x 次幂的项相加或相减。
- 将所有项移到一边：确保右边等于零。
3. Factorisation Method | 因式分解法
If the quadratic expression can be factorised into two linear factors, we can solve it using the zero product property. The method works when ax² + bx + c can be written as (px + q)(rx + s).
如果二次表达式可以因式分解为两个一次因式，我们可以利用零乘积性质来求解。当 ax² + bx + c 可以写成 (px + q)(rx + s) 时，这个方法有效。
For the simple case where a = 1, look for two numbers that multiply to give c and add to give b. For example, solve x² - 5x + 6 = 0:
对于 a = 1 的简单情况，找两个数，它们相乘等于 c，相加等于 b。例如，解 x² - 5x + 6 = 0：
x² - 5x + 6 = (x - 2)(x - 3) = 0
Then set each factor to zero:
然后将每个因式设为零：
x - 2 = 0 or x - 3 = 0
So x = 2 or x = 3. Always verify by substituting back into the original equation.
所以 x = 2 或 x = 3。务必通过代回原方程进行验证。
When a ≠ 1, use the method of grouping or trial and error. For example, solve 2x² + 7x + 3 = 0:
当 a ≠ 1 时，使用分组法或试错法。例如，解 2x² + 7x + 3 = 0：
(2x + 1)(x + 3) = 0
Then 2x + 1 = 0 gives x = -½, and x + 3 = 0 gives x = -3.
然后 2x + 1 = 0 得 x = -½，x + 3 = 0 得 x = -3。
4. Quadratic Formula | 求根公式
When factorisation is difficult or impossible, the quadratic formula solves any quadratic equation:
当因式分解困难或不可能时，求根公式可以解任何二次方程：
x = (−b ± √b² − 4ac) / (2a)
This formula is derived from completing the square and works for all a, b, c as long as a ≠ 0. Let's use it on 2x² - 4x - 3 = 0:
该公式由配方法推导而来，适用于所有 a、b、c，只要 a ≠ 0。我们用它在 2x² - 4x - 3 = 0 上：
Here a = 2, b = -4, c = -3. Substitute into the formula:
这里 a = 2，b = -4，c = -3。代入公式：
x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4
Thus the two solutions are x = (2 + √10)/2 and x = (2 - √10)/2. This method always gives exact solutions, even when they are irrational.
因此两个解是 x = (2 + √10)/2 和 x = (2 - √10)/2。这种方法总是给出精确的解，即使它们是无理数。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + h)² + k. This technique is useful for finding turning points and solving equations.
配方法将二次式改写为 a(x + h)² + k 的形式。这种技巧有助于找到转折点和解方程。
For a quadratic x² + bx + c, add and subtract (b/2)². Example: solve x² + 6x + 4 = 0 by completing the square.
对于二次式 x² + bx + c，加上并减去 (b/2)²。示例：通过配方法解 x² + 6x + 4 = 0。
First halve the coefficient of x: 6/2 = 3. Square it: 3² = 9. Rewrite the equation as:
首先将 x 的系数减半：6/2 = 3。平方：3² = 9。将方程改写为：
x² + 6x + 9 − 9 + 4 = 0
This gives (x + 3)² − 5 = 0. Hence (x + 3)² = 5. Taking square roots:
这得到 (x + 3)² − 5 = 0。因此 (x + 3)² = 5。开平方：
x + 3 = ±√5, so x = −3 ± √5
This method is especially valuable when the quadratic cannot be factorised easily.
当二次式不易因式分解时，这种方法尤为有价值。
6. The Discriminant | 判别式
The expression b² − 4ac inside the quadratic formula is called the discriminant, often denoted by Δ. It tells us the nature of the roots without solving the equation.
求根公式中的表达式 b² − 4ac 称为判别式，通常用 Δ 表示。它不需要解方程就能告诉我们根的性质。
| Discriminant | Nature of roots |
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root (equal roots) |
| Δ < 0 | No real roots (complex roots) |
判别式
| 判别式 | 根的性质 |
| Δ > 0 | 两个不相等的实数根 |
| Δ = 0 | 一个重复的实数根（相等根） |
| Δ < 0 | 没有实数根（复数根） |
For example, consider x² - 4x + 4 = 0. Here Δ = 16 - 16 = 0, so there is exactly one root: x = 2.
例如，考虑 x² - 4x + 4 = 0。这里 Δ = 16 - 16 = 0，所以恰好有一个根：x = 2。
If a question asks you to "show that the equation has no real roots", simply calculate Δ and show it is negative.
如果题目要求你“证明方程没有实数根”，只需计算 Δ 并证明它为负。
7. Graphs of Quadratic Functions | 二次函数图像
The graph of y = ax² + bx + c is a parabola. The sign of a determines its orientation:
函数 y = ax² + bx + c 的图像是抛物线。a 的符号决定其开口方向：
- 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，且有最小值点。
- 如果 a < 0，抛物线开口向下，且有最大值点。
The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the graph. The y-intercept is simply c. The vertex is the point where the curve changes direction; its x-coordinate is given by:
方程 ax² + bx + c = 0 的根对应图像与 x 轴的交点。y 截距就是 c。顶点是曲线改变方向的点；其 x 坐标为：
x = −b/(2a)
Then the y-coordinate can be found by substituting this x-value back into the original equation.
然后将这个 x 值代回原方程即可求出 y 坐标。
For example, for y = x² - 2x - 3, the vertex x-coordinate is 2/2 = 1. Substituting x = 1 gives y = 1 - 2 - 3 = -4. So the vertex is (1, -4).
例如，对于 y = x² - 2x - 3，顶点的 x 坐标为 2/2 = 1。代入 x = 1 得 y = 1 - 2 - 3 = -4。因此顶点是 (1, -4)。
8. Sum and Product of Roots | 根与系数的关系
For a quadratic equation ax² + bx + c = 0 with roots α and β, the following relationships hold:
对于具有根 α 和 β 的二次方程 ax² + bx + c = 0，以下关系成立：
Sum of roots: α + β = −b/a
Product of roots: αβ = c/a
These are extremely useful for checking answers or constructing equations with given roots.
这些关系对于检查答案或构造具有给定根的方程非常有用。
For example, if the roots of 2x² + 8x + 6 = 0 are α and β, then α + β = -4 and α β = 3. Indeed, factorising 2(x+1)(x+3)=0 gives roots -1 and -3, and their sum is -4, product is 3.
例如，如果 2x² + 8x + 6 = 0 的根是 α 和 β，则 α + β = -4，α β = 3。事实上，因式分解 2(x+1)(x+3)=0 得到根 -1 和 -3，其和为 -4，积为 3。
To form a new quadratic equation whose roots are α and β, use x² - (α + β)x + α β = 0.
若要构造一个根为 α 和 β 的新二次方程，使用 x² - (α + β)x + α β = 0。
9. Word Problems | 应用题
Quadratic equations often model real-life situations involving areas, projectile motion, or numbers. The key steps are:
二次方程常用于模拟涉及面积、抛体运动或数字的现实情境。关键步骤如下：
- Define the unknown variable.
- Translate the problem into an equation.
- Solve the equation using any suitable method.
- Interpret the solutions in the context of the problem.
- 定义未知变量。
- 将问题转化为方程。
- 使用任何合适的方法解方程。
- 在问题语境中解释解。
Example: A rectangular garden has length 4 metres more than its width. If its area is 45 m², find its dimensions.
示例：一个矩形花园的长度比宽度多 4 米。如果其面积为 45 平方米，求其尺寸。
Let the width be x metres. Then the length is x + 4. The area is x(x + 4) = 45, so:
设宽度为 x 米。则长度为 x + 4。面积为 x(x + 4) = 45，所以：
x² + 4x − 45 = 0
Factorise: (x + 9)(x − 5) = 0. Hence x = -9 or x = 5. Since width cannot be negative, x = 5. The length is 9 m.
因式分解：(x + 9)(x − 5) = 0。因此 x = -9 或 x = 5。由于宽度不能为负，x = 5。长度为 9 米。
Always discard solutions that do not make sense in the context.
始终舍弃在上下文中不合理的解。
10. Practice Questions | 练习
To master quadratic equations, practice is essential. Try these questions by yourself:
要掌握二次方程，练习必不可少。请自行尝试以下问题：
- Solve x² - 7x + 10 = 0.
- Solve 3x² + 5x - 2 = 0 using the quadratic formula.
- Find the roots of x² + 4x + 7 = 0. What does the discriminant tell you?
- Write x² + 6x - 1 in completed square form.
- The sum of the squares of two consecutive integers is 365. Find the integers.
- 解 x² - 7x + 10 = 0。
- 使用求根公式解 3x² + 5x - 2 = 0。
- 求 x² + 4x + 7 = 0 的根。判别式告诉你什么？
- 将 x² + 6x - 1 写成配方法形式。
- 两个连续整数的平方和为 365。求这两个整数。
Check your answers using the sum and product of roots. If you get stuck, go back to the relevant section.
利用根与系数的关系检查答案。如果卡住了，返回相关章节。
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