📚 Solving Quadratic Equations | 一元二次方程的求解
Quadratic equations appear throughout the IGCSE Mathematics syllabus. Mastering their solution methods is essential for success in both non-calculator and calculator papers. This revision guide covers the definition, standard forms, solution techniques, and common pitfalls.
一元二次方程贯穿整个 IGCSE 数学大纲。掌握其求解方法对于非计算器试卷和计算器试卷都至关重要。本复习指南将涵盖定义、标准形式、求解技巧以及常见易错点。
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. Its highest power of the unknown variable is two. The general form is written as ax2 + bx + c = 0 — but since we do not use LaTeX here, we write it as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
一元二次方程是次数为 2 的多项式方程，其未知数的最高次数为二。一般形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。
a = 0, the equation becomes linear, not quadratic. So the condition a ≠ 0 is fundamental.
如果 a = 0，方程变为一次方程，而非二次方程。因此 a ≠ 0 这一条件至关重要。
2. The General Form and Standard Form | 一般形式与标准形式
Any quadratic equation can be rearranged into the standard form ax² + bx + c = 0. For example, 2x2 = 3x - 1 becomes 2x2 - 3x + 1 = 0 .
任何一元二次方程都可以整理成标准形式 ax² + bx + c = 0。例如，2x² = 3x - 1 可化为 2x² - 3x + 1 = 0。
When solving, always write the equation in this form first. This avoids sign errors and makes the coefficients easy to read.
求解时，应先将方程写成这种形式。这样可以避免符号错误，并让系数一目了然。
ax² + bx + c = 0
3. Solving by Factorisation | 因式分解法
quadratic can be factorised into two linear brackets. For example, solve x² - 5x + 6 = 0.
当二次式可以分解成两个一次因式时，因式分解法是最快捷的方法。例如，解 x² - 5x + 6 = 0。
We look for two numbers that multiply to give +6 and add to give -5. Those numbers are -2 and -3. Hence:
我们寻找两个数，它们相乘等于 +6，相加等于 -5。这两个数是 -2 和 -3。因此：
(x - 2)(x - 3) = 0
zero, 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，则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。
> 1, the process is slightly longer. Solve 2x² + 5x - 3 = 0 by splitting the middle term.
当二次项系数 a > 1 时，过程稍长。用拆中项法解 2x² + 5x - 3 = 0。
- Multiply a and c: 2 × (-3) = -6.
- Find factors of -6 that add to +5: -1 and +6.
- Rewrite the middle term: 2x² - x + 6x - 3.
- Factor by grouping: x(2x - 1) + 3(2x - 1).
- Final factorisation: (2x - 1)(x + 3) = 0.
- 将 a 和 c 相乘：2 × (-3) = -6。
- 找到 -6 的两个因数，使其和等于 +5：-1 和 +6。
- 重写中间项：2x² - x + 6x - 3。
- 分组分解：x(2x - 1) + 3(2x - 1)。
- 最终分解：(2x - 1)(x + 3) = 0。
Therefore x = ½ or x = -3.
因此 x = ½ 或 x = -3。
4. The Quadratic Formula | 求根公式
The quadratic formula works for any quadratic equation, even when factorisation is difficult or impossible. The formula is:
求根公式适用于任何一元二次方程，即使在分解困难或无法分解的情况下也有效。公式如下：
x = (-b ± √b² - 4ac) / (2a)
symbol ± means that there are two values: one with a plus sign and one with a minus sign.
这里的 ± 符号表示有两个值：一个取加号，一个取减号。
Example: Solve 3x² - 4x - 2 = 0 using the formula. Here a = 3, b = -4, c = -2.
例如：用求根公式解 3x² - 4x - 2 = 0。此时 a = 3，b = -4，c = -2。
x = (4 ± √((-4)² - 4 × 3 × (-2))) / (2 × 3)
x = (4 ± √16 + 24) / 6 = (4 ± √40) / 6
Since √40 = 2√10, the solutions are x = (4 + 2√10)/6 and x = (4 - 2√10)/6. These can be simplified to x = (2 ± √10)/3.
由于 √40 = 2√10，解为 x = (4 + 2√10)/6 和 x = (4 - 2√10)/6。可化简为 x = (2 ± √10)/3。
Always check whether you are asked to give exact answers or answers correct to a certain number of decimal places.
注意题目要求精确答案还是保留到小数位的近似答案。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful for finding turning points and solving equations without using the formula.
配方法将二次式改写为 a(x + p)² + q 的形式。这有助于求顶点坐标，也可用于解方程而不使用求根公式。
Example: Write x² + 6x + 5 in the form (x + p)² + q.
例如：将 x² + 6x + 5 改写为 (x + p)² + q 的形式。
Take half of 6, which is 3, and square it to get 9. So:
取 6 的一半，即 3，平方得 9。因此：
x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4
)² - 4 = 0, add 4 to both sides and take the square root: x + 3 = ±2, so x = -1 or x = -5.
要解 (x + 3)² - 4 = 0，两边同时加 4，再开平方：x + 3 = ±2，因此 x = -1 或 x = -5。
If the coefficient a is not 1, factorise it out first before completing the square.
如果二次项系数 a 不是 1，需要先提取公因数，再进行配方。
6. The Discriminant and the Nature of Roots | 判别式与根的性质
The expression b² - 4ac is called the discriminant. It tells us how many real roots a quadratic equation has.
代数式 b² - 4ac 称为判别式。它告诉我们一元二次方程有多少个实数根。
| Value of b² - 4ac | Nature of roots |
|---|---|
| Positive and a perfect square | Two distinct rational roots |
| Positive but not a perfect square | Two distinct irrational roots |
| Zero | Two equal real roots (one repeated root) |
| Negative | No real roots |
判别式 b² - 4ac 的值为正且是完全平方数时，方程有两个不相等的有理数根；为正但不是完全平方数时，方程有两个不相等的无理数根；为零时，方程有两个相等的实数根（一个重根）；为负时，方程没有实数根。
x + 1 = 0, find the set of values of k for which the equation has two distinct real roots.
例如：对于 kx² + 4x + 1 = 0，求 k 的取值范围，使方程有两个不相等的实数根。
4² - 4 × k × 1 > 0
16 - 4k > 0
So k < 4, but since the equation is quadratic we also need k ≠ 0. Therefore the set of values is k < 4 and k ≠ 0.
所以 k < 4，但因为是二次方程，k 还必须满足 k ≠ 0。因此取值范围为 k < 4 且 k ≠ 0。
7. Sum and Product of Roots | 根与系数的关系
r a quadratic equation ax² + bx + c = 0 with roots α and β (alpha and beta), we can write:
对于二次方程 ax² + bx + c = 0，设其两根为 α 和 β，则有：
Sum of roots: α + β = -b/a
Product of roots: α × β = c/a
Example: A quadratic equation has roots 3 and -5. Find the equation.
例如：已知某二次方程的两根为 3 和 -5，求该方程。
5, so c/a = -15. Choosing a = 1 gives b = 2 and c = -15. The equation is x² + 2x - 15 = 0.
两根和 = 3 + (-5) = -2，所以 -b/a = -2。两根积 = 3 × (-5) = -15，所以 c/a = -15。令 a = 1，得 b = 2，c = -15。方程为 x² + 2x - 15 = 0。
This relationship is particularly useful in questions that give the roots without providing the equation itself.
这种关系特别适用于题目直接给出两根而要求构造方程的题型。
8. Graphs of Quadratic Functions | 二次函数图像
bx + c is a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上；当 a < 0 时，抛物线开口向下。
ph are the real roots of the equation ax² + bx + c = 0. The y-intercept is the value of c.
抛物线与 x 轴的交点就是方程 ax² + bx + c = 0 的实数根。与 y 轴的交点为 (0, c)。
The vertex (turning point) has x-coordinate given by x = -b/(2a). You can then substitute this value back into the equation to find the y-coordinate.
顶点（转折点）的横坐标为 x = -b/(2a)，再将该横坐标代入原式即可求出纵坐标。
ation is written in completed square form y = a(x + p)² + q, the vertex is simply (-p, q).
若方程已写成配方法形式 y = a(x + p)² + q，则顶点坐标直接为 (-p, q)。
9. Word Problems Leading to Quadratics | 二次方程应用题
Many exam problems require you to set up a quadratic equation from a real-world situation. Common contexts include areas, product of consecutive numbers, and projectile motion.
许多考试题需要你根据现实情境建立二次方程。常见背景包括面积、连续整数乘积和抛体运动。
Example: The length of a rectangle is 3 cm longer than its width. Its area is 40 cm². Find the width.
例如：一个长方形的长比宽多 3 cm，面积为 40 cm²。求宽。
Let the width be x cm. Then the length is x + 3 cm. Area = x(x + 3) = 40.
设宽为 x cm，则长为 x + 3 cm。面积 = x(x + 3) = 40。
x² + 3x - 40 = 0
actorise: (x + 8)(x - 5) = 0. Since width cannot be negative, x = 5. So the width is 5 cm.
分解因式：(x + 8)(x - 5) = 0。因为宽不能为负数，所以 x = 5，即宽为 5 cm。
Always reject negative solutions when the variable represents a physical length or a count.
当变量表示实际长度或数量时，一定要舍去负数解。
10. Common Mistakes | 常见错误
Here are frequent errors that cost marks in exams. Avoid them by practising carefully.
以下是在考试中常见的失分错误。通过仔细练习来避免它们。
- Forgetting to rearrange the equation into standard form before factorising or using the formula.
- Dropping the negative sign when substituting b into the quadratic formula.
- Writing x = ±√k without first isolating the squared term.
- Dividing both sides of an equation by x when x may equal zero, thus losing a root.
- Confusing the sum and product of roots formulas.
- 在分解或使用求根公式前，忘记将方程整理为标准形式。
- 在代入 b 时漏掉负号。
- 没有先分离平方项就写出 x = ±√k。
- 在 x 可能等于零的情况下两边同时除以 x，导致丢根。
- 混淆两根之和与两根之积的公式。
Check your answers by substituting them back into the original equation. This only takes a few seconds and catches most errors.
将解代回原方程进行验算。这只需几秒钟，却能发现大部分错误。
11. Exam Tips for IGCSE | IGCSE 考试技巧
In the non-calculator paper, factorisation and completing the square are often expected. In the calculator paper, the quadratic formula can be used freely, but show all working clearly.
在非计算器试卷中，通常期望使用因式分解法和配方法。在计算器试卷中，可以自由使用求根公式，但必须写出清晰的解题过程。
If a question asks for answers "correct to 2 decimal places", use the quadratic formula and round at the very end. If it asks for "exact values", use surds or factorisation.
如果题目要求"保留两位小数"，应使用求根公式并在最后一步四舍五入。如果要求"精确值"，则使用根式或因式分解。
Read the question carefully to decide how many solutions are needed. Sometimes only positive values are meaningful.
仔细审题以确定需要多少个解。有时只有正数解才具有实际意义。
Remember that the graph of a quadratic function can help you estimate the number of real roots before you solve. Use this as a sanity check.
记住，二次函数图像可以帮助你在求解前估计实数根的个数。可以用它来检验答案是否合理。
12. Practice Questions | 练习题目
Attempt these questions on your own, then check your answers by substituting back.
请独立尝试以下题目，然后通过代回验算检查答案。
- Solve x² - 7x + 10 = 0.
- Solve 2x² + x - 6 = 0.
- Solve x² + 8x + 2 = 0 by completing the square.
- e quadratic formula to solve 5x² - 3x - 1 = 0, giving answers correct to 2 decimal places.
- Find the value(s) of k such that x² + kx + 9 = 0 has two equal roots.
- 解方程 x² - 7x + 10 = 0。
- 解方程 2x² + x - 6 = 0。
- 用配方法解 x² + 8x + 2 = 0。
- 用求根公式解 5x² - 3x - 1 = 0，答案保留两位小数。
- 求 k 的值，使 x² + kx + 9 = 0 有两个相等的实数根。
Answers: 1) x = 2 or 5. 2) x = 3/2 or -2. 3) x = -4 ± √14. 4) x ≈ 0.87 or -0.23. 5) k = 6 or -6.
答案：1) x = 2 或 5。2) x = 3/2 或 -2。3) x = -4 ± √14。4) x ≈ 0.87 或 -0.23。5) k = 6 或 -6。
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