📚 Solving Quadratic Equations | 二次方程求解
Quadratic equations are one of the central topics in IGCSE Mathematics. They appear in algebra, geometry, coordinate geometry, and word problems. A confident command of quadratic methods will help you solve many different question types accurately.
二次方程是 IGCSE 数学的核心主题之一。它们出现在代数、几何、坐标几何和应用题中。熟练掌握二次方程的各种解法，将帮助你准确解答许多不同类型的题目。
1. What is a Quadratic Equation? | 什么是二次方程？
0, where a, b and c are constants and a cannot be zero. If a were zero, the equation would become linear rather than quadratic.
二次方程是未知数最高次数为 2 的方程。一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a 不能为零。如果 a 为零，方程就会变成一次方程，而不是二次方程。
For IGCSE, you will meet quadratic equations in pure algebra questions, in geometry problems such as finding unknown sides of rectangles, and in number puzzles involving consecutive integers.
在 IGCSE 数学中，你会在纯代数题、几何问题（如求矩形的未知边长）以及涉及连续整数的数字谜题中遇到二次方程。
2. Standard Form and Coefficients | 标准形式与系数
form ax² + bx + c = 0. Collect all terms on one side and set the other side equal to zero.
在选择解题方法之前，始终将二次方程整理为标准形式 ax² + bx + c = 0。将所有项移到一边，并令另一边等于零。
+ 3 = 0, we have a = 2, b = −5 and c = 3. In x² − 4 = 0, we have a = 1, b = 0 and c = −4.
仔细识别 a、b、c，包括负号。例如，在 2x² − 5x + 3 = 0 中，a = 2，b = −5，c = 3。在 x² − 4 = 0 中，a = 1，b = 0，c = −4。
Writing the equation in standard form first reduces mistakes and helps you select the most efficient method.
先将方程写成标准形式可以减少错误，并帮助你选择最有效的解法。
3. Solving by Factorising | 因式分解法
When a quadratic expression factorises easily, factorising is usually the fastest method. We rewrite ax² + bx + c as a product of two linear brackets.
当二次表达式容易因式分解时，因式分解法通常是最快的方法。我们将 ax² + bx + c 写成两个一次括号的乘积。
= 0 becomes (x + 2)(x + 3) = 0. The numbers 2 and 3 add to give 5 and multiply to give 6.
例如，x² + 5x + 6 = 0 可写成 (x + 2)(x + 3) = 0。数字 2 和 3 相加得 5，相乘得 6。
x² + 5x + 6 = (x + 2)(x + 3)
nd 1, so the expression becomes 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
当 a ≠ 1 时，例如 2x² + 7x + 3，寻找两个数，它们相乘得 a × c = 6，相加得 b = 7。这两个数是 6 和 1，所以表达式变为 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
4. The Zero Product Property | 零乘积性质
Once the quadratic is written as a product of brackets, use the zero product property: if the product of two expressions is zero, then at least one of the expressions must be zero.
当二次方程写成括号乘积后，使用零乘积性质：如果两个表达式的乘积为零，那么其中至少一个表达式必须为零。
the resulting linear equations. For (x + 2)(x + 3) = 0, we obtain x + 2 = 0 or x + 3 = 0.
令每个括号等于零，然后解所得的一次方程。对于 (x + 2)(x + 3) = 0，我们得到 x + 2 = 0 或 x + 3 = 0。
(x + 2)(x + 3) = 0
⇒ x + 2 = 0 or x + 3 = 0
⇒ x = −2 or x = −3
Always write both possible roots clearly. IGCSE mark schemes often require both solutions unless the question states otherwise.
始终清晰地写出两个可能的根。除非题目另有说明，IGCSE 评分标准通常要求写出两个解。
5. Completing the Square | 配方法
uadratic does not factorise easily and when you need to find the vertex of a quadratic graph.
配方法将二次方程转化为 (x + p)² + q = 0 的形式。当二次方程不容易因式分解时，以及当你需要求二次图像顶点时，这个方法非常有用。
(b/2)². In words, take half the coefficient of x, square it, and adjust by subtracting that square.
关键恒等式是 x² + bx = (x + b/2)² − (b/2)²。也就是说，取 x 系数的一半，平方，并减去这个平方数进行调整。
, complete the square for x² + 6x. Half of 6 is 3, and 3² is 9, so x² + 6x = (x + 3)² − 9.
例如，解 x² + 6x + 1 = 0 时，对 x² + 6x 进行配方。6 的一半是 3，3² 是 9，所以 x² + 6x = (x + 3)² − 9。
x² + 6x + 1 = 0
⇒ (x + 3)² − 9 + 1 = 0
⇒ (x + 3)² = 8
⇒ x + 3 = ±√8
⇒ x = −3 ± 2√2
Notice that answers may be left in surd form unless the question asks for decimals.
注意，除非题目要求小数，否则答案可以用根式形式表示。
6. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation in standard form. It is especially useful when factorising is difficult, when coefficients are not integers, or when a calculator is allowed.
求根公式可以解任何标准形式的二次方程。当因式分解困难、系数不是整数或允许使用计算器时，它尤其有用。
For a quadratic equation ax² + bx + c = 0, the solutions are given by:
对于二次方程 ax² + bx + c = 0，其解由以下公式给出：
x = ( −b ± √b² − 4ac ) / (2a)
Always write the formula first, then substitute the values of a, b and c. Simplify inside the square root before dividing by 2a.
始终先写出公式，然后代入 a、b、c 的值。先化简根号内部，再除以 2a。
For example, in 2x² − 3x − 4 = 0, a = 2, b = −3 and c = −4. Substitution gives:
例如，在 2x² − 3x − 4 = 0 中，a = 2，b = −3，c = −4。代入可得：
x = ( −(−3) ± √((−3)² − 4(2)(−4)) ) / (2(2))
⇒ x = ( 3 ± √9 + 32 ) / 4
⇒ x = ( 3 ± √41 ) / 4
7. Discriminant and Nature of Roots | 判别式与根的性质
imes written as Δ. It tells us the nature of the roots without solving the equation completely.
判别式是 D = b² − 4ac，有时写作 Δ。它可以在不完整解方程的情况下告诉我们根的性质。
al roots. If D = 0, there is one repeated real root. If D < 0, there are no real roots.
如果 D > 0，方程有两个不同的实根；如果 D = 0，有一个重根；如果 D < 0，没有实根。
| Discriminant | Nature of roots |
| b² − 4ac > 0 | Two distinct real roots |
| b² − 4ac = 0 | One repeated real root |
| b² − 4ac < 0 | No real roots |
IGCSE questions often ask you to find the value of a constant for which a quadratic has equal roots. In that case, set b² − 4ac = 0.
IGCSE 题目经常要求你求出使二次方程有两个相等实根的常数值。这时令 b² − 4ac = 0 即可。
8. Graphical Interpretation | 图像解释
he solutions of ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.
y = ax² + bx + c 的图像是抛物线。方程 ax² + bx + c = 0 的解是图像与 x 轴交点的 x 坐标。
If the discriminant is positive, the parabola crosses the x-axis twice. If it is zero, the parabola touches the x-axis once. If it is negative, the parabola never crosses the x-axis.
如果判别式为正，抛物线与 x 轴有两个交点；如果为零，抛物线与 x 轴相切于一点；如果为负，抛物线不与 x 轴相交。
q is (−p, q). Completing the square is the link between the algebraic solution and the graph.
抛物线 y = (x + p)² + q 的顶点是 (−p, q)。配方法将代数解与图像联系起来。
9. Word Problems and Applications | 应用题
Many IGCSE word problems lead to a quadratic equation. Define the unknown clearly, form an equation, solve it, and reject any solution that does not make sense in the context.
许多 IGCSE 应用题会导出一个二次方程。清晰地设未知数，建立方程，解方程，并舍去在实际情况中不合理的解。
tegers is 56, let the integers be n and n + 1. Then n(n + 1) = 56, giving n² + n − 56 = 0.
例如，两个连续正整数的积为 56，设这两个整数为 n 和 n + 1。则 n(n + 1) = 56，得到 n² + n − 56 = 0。
n² + n − 56 = 0
⇒ (n + 8)(n − 7) = 0
⇒ n = 7 (since n is positive)
Negative solutions are often mathematically valid but must be rejected in measurement or counting problems. Always check the physical meaning of your answer.
负解在数学上往往有效，但在测量或计数问题中必须舍去。始终检查答案的实际意义。
10. Common Mistakes | 常见错误
One common mistake is forgetting to set the equation equal to zero before factorising. If you factorise without rearranging first, the zero product property cannot be applied.
一个常见错误是在因式分解前忘记将方程移项使其等于零。如果不先移项就因式分解，就无法应用零乘积性质。
hich can lose the root x = 0. Instead, bring all terms to one side and factorise properly.
另一个错误是将两边同时除以 x，这可能会丢失 x = 0 这个根。正确做法是将所有项移到一边并正确因式分解。
Be careful with signs when substituting into the quadratic formula, especially when b is negative or c is negative. Use brackets around negative values when using a calculator.
在代入求根公式时要仔细检查符号，尤其是当 b 为负数或 c 为负数时。使用计算器时应对负数加括号。
11. Exam Tips | 考试技巧
In the IGCSE exam, show all steps: standard form, choice of method, substitution, simplification, and final answers. Clear working can earn method marks even if the final answer is wrong.
在 IGCSE 考试中，要展示所有步骤：标准形式、方法选择、代入、化简和最终答案。清晰的解题过程
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