📚 Solving Quadratic Equations | 解一元二次方程
Quadratic equations appear in nearly every IGCSE Mathematics paper. Whether you are factorising, completing the square, or using the formula, a clear understanding of the logic behind each method is essential for full marks.
一元二次方程幾乎出现在每份 IGCSE 数学试卷中。无论是因式分解、配方法还是求根公式，理解每种方法背后的逻辑都是拿到满分的关键。
1. What Is a Quadratic Equation? | 什么是一元二次方程？
A quadratic equation is a polynomial equation in which the highest power of the unknown is 2. Its graph is always a smooth curve called a parabola.
一元二次方程是未知数最高次数为 2 的多项式方程，其图像总是一条平滑的曲线，称为抛物线。
In its most useful form, a quadratic equation is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is vital: if a = 0, the equation becomes linear.
在最常用的形式中，一元二次方程写作 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。条件 a ≠ 0 至关重要：若 a = 0，方程就变成了线性方程。
For example, x² − 5x + 6 = 0 and 2x² + 3x − 1 = 0 are quadratics, while x³ − 1 = 0 is not, because its highest power is 3.
例如，x² − 5x + 6 = 0 和 2x² + 3x − 1 = 0 是二次方程，而 x³ − 1 = 0 不是，因为它的最高次数为 3。
2. Standard Form and Key Terms | 标准形式与关键术语
Before solving, always rearrange the equation into the standard form ax² + bx + c = 0 on one side, with zero on the other side.
在求解之前，务必把方程整理成标准形式 ax² + bx + c = 0，即一边为多项式，另一边为零。
The values a and b are called coefficients, and c is the constant term. The solutions of the equation are called the roots or x-intercepts; they are the x-values where the graph crosses the x-axis.
a 和 b 称为系数，c 称为常数项。方程的解称为根或 x 轴截距，它们是图像与 x 轴交点的 x 坐标。
One key idea used in factorisation is the zero product property: if p × q = 0, then p = 0 or q = 0. This property allows us to split a quadratic into two simple linear equations.
因式分解中的关键思想是零乘积性质：若 p × q = 0，则 p = 0 或 q = 0。利用这个性质，我们可以把一个二次方程拆成两个简单的一次方程。
3. Method 1: Factorisation | 方法一：因式分解法
Factorisation is fastest when a, b and c are integers and the quadratic can be written as a product of two linear brackets.
当 a、b、c 都是整数且二次式能写成两个一次括号的乘积时，因式分解法最快。
For the simple case x² + bx + c = 0, look for two numbers that multiply to give c and add to give b.
对于简单情形 x² + bx + c = 0，寻找两个数，使它们的乘积等于 c，且它们的和等于 b。
Example: solve x² − 5x + 6 = 0. We need two numbers whose product is 6 and whose sum is −5. These are −2 and −3.
例：解 x² − 5x + 6 = 0。我们需要两个数，其乘积为 6，和为 −5。这两个数是 −2 和 −3。
(x − 2)(x − 3) = 0
By the zero product property, x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.
由零乘积性质，x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。
When a ≠ 1, use the product-and-sum method in pairs. Example: solve 2x² + 5x − 3 = 0. Multiply a and c: 2 × (−3) = −6. Find two numbers with product −6 and sum 5: they are 6 and −1. Then split the middle term:
当 a ≠ 1 时，可使用分组法。例：解 2x² + 5x − 3 = 0。先计算 a 与 c 的乘积：2 × (−3) = −6。找出乘积为 −6、和为 5 的两个数：它们是 6 和 −1。然后拆分中间项：
2x² + 6x − x − 3 = 0
Group terms: 2x(x + 3) − 1(x + 3) = 0, so (2x − 1)(x + 3) = 0. Hence x = 1/2 or x = −3. Always check your brackets by expanding them.
分组：2x(x + 3) − 1(x + 3) = 0，即 (2x − 1)(x + 3) = 0。因此 x = 1/2 或 x = −3。务必通过展开括号来检验。
4. Method 2: Completing the Square | 方法二：配方法
Completing the square rewrites x² + bx + c as (x + p)² + q. This form reveals the turning point of the parabola directly.
配方法将 x² + bx + c 改写为 (x + p)² + q 的形式，这种形式能直接显示抛物线的顶点坐标。
The procedure: take half of b, write (x + b/2)², then subtract (b/2)² and add c.
操作步骤：取 b 的一半，写出 (x + b/2)²，然后减去 (b/2)²，再加上 c。
x² + bx + c = (x + b/2)² − (b/2)² + c
Example: solve x² + 6x − 7 = 0. Half of 6 is 3, so (x + 3)² − 9 − 7 = 0, hence (x + 3)² − 16 = 0.
例：解 x² + 6x − 7 = 0。6 的一半是 3，所以 (x + 3)² − 9 − 7 = 0，即 (x + 3)² − 16 = 0。
Rearrange: (x + 3)² = 16. Taking square roots gives x + 3 = ±4, so x = 1 or x = −7.
整理得 (x + 3)² = 16。两边开平方得 x + 3 = ±4，所以 x = 1 或 x = −7。
If a ≠ 1, divide the whole equation by a first. Completing the square is especially useful when the coefficient of x is even, because the arithmetic stays clean.
如果 a ≠ 1，先把整个方程除以 a。当 x 的系数为偶数时，配方法尤其方便，因为计算过程更整洁。
5. Method 3: The Quadratic Formula | 方法三：求根公式
The quadratic formula solves any quadratic equation, including those that cannot be factorised easily. For ax² + bx + c = 0, the roots are given by:
求根公式可以解任何一元二次方程，包括那些难以因式分解的方程。对于 ax² + bx + c = 0，根为：
x = [−b ± √(b² − 4ac)] ÷ 2a
Example: solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute carefully:
例：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入时务必小心：
x = [4 ± √(16 + 24)] ÷ 4 = (4 ± √40) ÷ 4
Simplify the surd: √40 = 2√10, so x = (2 ± √10) ÷ 2. As decimals, x ≈ 2.58 or x ≈ −0.58 (to 2 decimal places).
化简根式：√40 = 2√10，所以 x = (2 ± √10) ÷ 2。化为小数得 x ≈ 2.58 或 x ≈ −0.58（精确到两位小数）。
When using the formula, write down the values of a, b and c first, and substitute them in one step to avoid sign errors.
使用求根公式时，先写出 a、b、c 的数值，然后一次性代入，这样可以避免符号错误。
6. The Discriminant Δ | 判别式 Δ
The expression b² − 4ac inside the square root is called the discriminant, written as Δ. It tells us about the nature of the roots without solving the equation fully.
根号内的表达式 b² − 4ac 称为判别式，记作 Δ。它无需完整求解就能告诉我们方程的根的性质。
| Δ = b² − 4ac | Nature of Roots | 根的性质 |
| Δ > 0 and a perfect square | Two distinct rational roots | 两个不等有理根 |
| Δ > 0, not a perfect square | Two distinct irrational roots | 两个不等无理根 |
| Δ = 0 | One repeated real root | 一个重根（两个相等实根） |
| Δ < 0 | No real roots | 无实根 |
Example: find the value of k such that kx² − 4x + 1 = 0 has exactly one repeated root. Set Δ = 0: (−4)² − 4 × k × 1 = 0, so 16 − 4k = 0, giving k = 4.
例：求 k 的值，使 kx² − 4x + 1 = 0 有且仅有一个重根。令 Δ = 0：(−4)² − 4 × k × 1 = 0，即 16 − 4k = 0，所以 k = 4。
Always consider Δ before choosing a method. If Δ is a negative number, you can immediately conclude there are no real solutions and stop.
在选择方法之前先计算 Δ。如果 Δ 为负数，你可以立即判断无实根并停止求解。
7. Solving by Graph | 图像法求解
Graphically, the roots of ax² + bx + c = 0 are the x-coordinates where the parabola y = ax² + bx + c crosses the x-axis.
从图像上看，ax² + bx + c = 0 的根就是抛物线 y = ax² + bx + c 与 x 轴交点的 x 坐标。
If the parabola touches the x-axis at one point, the equation has one repeated root; if it does not intersect the x-axis, there are no real roots.
若抛物线与 x 轴仅相切于一点，则方程有一个重根；若不相交，则无实根。
Completing the square is the fastest way to sketch a quadratic. From y = (x + p)² + q, the vertex is (−p, q). For example, y = (x − 2)² − 9 has vertex (2, −9) and roots at x = −1 and x = 5, since (x − 2)² = 9 gives x = 2 ± 3.
配方法是快速画出二次函数图像的最快捷方式。由 y = (x + p)² + q，顶点为 (−p, q)。例如，y = (x − 2)² − 9 的顶点为 (2, −9)，由 (x − 2)² = 9 得 x = 2 ± 3，所以根为 x = −1 和 x = 5。
8. Word Problems | 应用题
Many IGCSE questions require you to form a quadratic equation from a real-life situation and then solve it.
许多 IGCSE 题目要求你从实际情境中建立一元二次方程，然后求解。
Example: the product of two consecutive positive integers is 56. Let the smaller integer be x. Then x(x + 1) = 56.
例：两个连续正整数的乘积为 56。设较小的整数为 x，则 x(x + 1) = 56。
x² + x − 56 = 0
Factorise: (x + 8)(x − 7) = 0, so x = −8 or x = 7. Since the integers are positive, reject x = −8. The integers are 7 and 8.
因式分解：(x + 8)(x − 7) = 0，所以 x = −8 或 x = 7。由于整数为正数，舍去 x = −8。这两个数为 7 和 8。
Always define your variable clearly at the start, and check whether the solution makes sense in the context. A length or a count cannot be negative.
开始时务必清晰定义未知数，并检查解在情境中是否合理。长度或数量不能为负数。
9. Common Mistakes | 常见错误
One frequent error is trying to solve a quadratic before rearranging it to equal zero. For x² = 3x, students may cancel x and get only x = 3, losing the root x = 0. Correct working: x² − 3x = 0, so x(x − 3) = 0, giving x = 0 or x = 3.
一个常见错误是在未把方程整理成等于零之前就求解。例如 x² = 3x，学生可能直接约去 x 而只得到 x = 3，从而丢失根 x = 0。正确做法：x² − 3x = 0，即 x(x − 3) = 0，得 x = 0 或 x = 3。
Sign errors are common when substituting into the quadratic formula. Remember that −b changes the sign of b, and the discriminant must be simplified before taking its square root.
代入求根公式时常出现符号错误。请注意 −b 要改变 b 的符号，并且开平方前必须先化简判别式。
In completing the square, students often forget to subtract the square of half of b. For x² + 6x, the completed form is (x + 3)² − 9, not (x + 3)².
配方法中，学生常忘记减去 b 的一半的平方。对于 x² + 6x，配方式应为 (x + 3)² − 9，而不是 (x + 3)²。
Finally, never divide both sides by an unknown that could be zero. This destroys solutions and is a serious exam error.
最后，切勿将等式两边同时除以可能为零的未知数，这会丢失解，属于严重的考试错误。
10. Exam Tips | 考试技巧
Choose the quickest reliable method per question. For a simple integer quadratic, factorise; for awkward coefficients, use the formula; to sketch a graph, complete the square.
根据题目选择最快捷可靠的方法：简单整系数二次式用因式分解；系数复杂时用求根公式；画图时用配方法。
Always substitute your roots back into the original equation to check them. This takes ten seconds and catches most algebraic slips.
务必把根代回原方程检验。这只需十秒钟，却能捕捉到大多数代数疏漏。
If a question asks for answers to two decimal places, show your calculation using the formula first and round only at the final step.
如果题目要求精确到两位小数，先用公式写出计算过程，最后一步才进行四舍五入。
Look out for special patterns: x² − 9 = 0 factors as (x − 3)(x + 3) by the difference of squares, and x² − 6x + 9 = 0 is a perfect square (x − 3)² = 0.
留意特殊模式：x² − 9 = 0 可用平方差公式分解为 (x − 3)(x + 3)，而 x² − 6x + 9 = 0 是完全平方式 (x − 3)² = 0。
Practise past-paper questions repeatedly until factorisation becomes automatic. Speed and accuracy with quadratics will reward you across Paper 2 and Paper 4.
反复练习真题，直到因式分解变得熟练自如。快速而准确地处理二次方程，将在 Paper 2 和 Paper 4 中为你带来丰厚的回报。
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