📚 Mastering Quadratic Equations: A Complete Guide for IGCSE | 掌握一元二次方程：IGCSE 数学完整指南
Quadratic equations are a cornerstone of the IGCSE Mathematics curriculum. Whether you are solving for unknown roots, interpreting graphs, or applying them to real-world scenarios, a solid grasp of this topic is essential for success in both Paper 2 and Paper 4. This guide breaks down every key concept, method, and common pitfall you need to know.
一元二次方程是 IGCSE 数学课程的核心内容。无论你是要求解未知根、解读图像，还是将其应用于现实场景，牢固掌握这一主题对在 Paper 2 和 Paper 4 中取得高分都至关重要。本指南将系统地拆解每一个关键概念、解题方法和常见误区，助你全面备考。
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 standard form is written as ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The value of a cannot be zero because then the equation would become linear.
一元二次方程是次数为 2 的多项式方程。其标准形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。a 不能为零，否则方程将退化为一次方程。
When the equation is expressed in this form, the left-hand side is a quadratic expression, and the solutions to the equation are called the roots or solutions of the quadratic. These roots are the x-values where the corresponding quadratic graph y = ax² + bx + c crosses the x-axis.
当方程写作该形式时，左边是一个二次表达式，方程的解被称为该二次方程的根或解。这些根就是对应的抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。
2. Expanding and Factorising: The Foundation | 展开与因式分解：基础技能
Before solving quadratics, you must be comfortable expanding and factorising. Expanding means removing brackets, for example: (x + 3)(x + 5) = x² + 8x + 15. Factorising is the reverse process: x² + 8x + 15 = (x + 3)(x + 5).
在求解二次方程之前，你必须熟练展开和因式分解。展开意味着去掉括号，例如：(x + 3)(x + 5) = x² + 8x + 15。因式分解是反向过程：x² + 8x + 15 = (x + 3)(x + 5)。
For the general expansion (px + q)(rx + s), the result is prx² + (ps + qr)x + qs. Mastery of this skill is critical because the most efficient method of solving many quadratic equations relies on factorisation.
对于一般的展开式 (px + q)(rx + s)，结果为 prx² + (ps + qr)x + qs。掌握这一技能至关重要，因为解许多二次方程最有效的方法就是因式分解。
3. Method 1: Solving by Factorisation | 方法一：因式分解法
When a quadratic equation is set equal to zero, you can often factorise it into two linear factors. Consider the equation x² + 8x + 15 = 0. To factorise, find two numbers that multiply to 15 and add to 8 — those numbers are 3 and 5.
当二次方程等于零时，通常可以将其分解为两个一次因式。考虑方程 x² + 8x + 15 = 0。要因式分解，找到两个相乘为 15、相加为 8 的数——它们是 3 和 5。
Thus, the equation becomes (x + 3)(x + 5) = 0. Since the product of two expressions is zero, at least one of them must be zero. Therefore, x + 3 = 0 or x + 5 = 0, giving the roots x = -3 and x = -5.
因此，方程变为 (x + 3)(x + 5) = 0。由于两个表达式的乘积为零，至少其中一个必须为零。所以 x + 3 = 0 或 x + 5 = 0，从而得到根 x = -3 和 x = -5。
This method is fast but works only when the quadratic can be factorised cleanly. If a > 1, you need to find factors of ac that add up to b, then use grouping. For example, for 2x² – 7x + 3 = 0, ac = 6, and the two numbers are -6 and -1, since (-6) + (-1) = -7.
这种方法很快，但仅在二次方程能够干净地因式分解时有效。如果 a > 1，则需要找到 ac 的因子，使其相加等于 b，然后使用分组法。例如，对于 2x² – 7x + 3 = 0，ac = 6，两个数是 -6 和 -1，因为 (-6) + (-1) = -7。
4. Method 2: Completing the Square | 方法二：配方法
Completing the square is a powerful technique that rewrites a quadratic in the form a(x + p)² + q. This makes it easy to identify the vertex of the parabola and solve equations even when factorisation is not possible.
配方是一种强大的技巧，它将二次式重写为 a(x + p)² + q 的形式。这使我们可以轻松找到抛物线的顶点，并且即使无法因式分解也能求解方程。
For a simple example, consider x² + 6x + 2 = 0. Take half of the coefficient of x, which is 3, and square it to get 9. Rewrite the equation as (x + 3)² – 9 + 2 = 0, which simplifies to (x + 3)² = 7.
举一个简单的例子，考虑 x² + 6x + 2 = 0。取 x 系数的一半，即 3，然后平方得 9。将方程重写为 (x + 3)² – 9 + 2 = 0，即 (x + 3)² = 7。
Taking the square root of both sides gives x + 3 = ±√7, so x = -3 ± √7. There are always two solutions, one with the positive square root and one with the negative square root.
两边开平方得 x + 3 = ±√7，因此 x = -3 ± √7。总是有两个解，一个取正平方根，一个取负平方根。
5. Method 3: The Quadratic Formula | 方法三：求根公式法
The quadratic formula is the most general method. For any quadratic equation ax² + bx + c = 0, the roots are given by:
求根公式是最通用的方法。对于任意一元二次方程 ax² + bx + c = 0，其根为：
x = (-b ± √(b² – 4ac)) / (2a)
This formula works for all quadratics, including those with irrational roots. To use it, simply identify a, b, c from the equation, substitute into the formula, and simplify step by step.
该公式适用于所有二次方程，包括含有无理根的方程。使用时，只需从方程中找出 a、b、c，代入公式并逐步化简即可。
For example, for 2x² + 3x – 5 = 0, we have a = 2, b = 3, c = -5. Substituting gives x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4. Thus x = 1 or x = -2.5.
例如，对于 2x² + 3x – 5 = 0，有 a = 2，b = 3，c = -5。代入得 x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4。因此 x = 1 或 x = -2.5。
6. The Discriminant: b² – 4ac | 判别式：b² – 4ac
The discriminant is the expression under the square root in the quadratic formula: Δ = b² – 4ac. It tells us the nature of the roots without fully solving the equation.
判别式是求根公式中被开方的部分：Δ = b² – 4ac。它无需完整解方程即可告诉我们根的性质。
If Δ > 0, the equation has two real and distinct roots. If Δ = 0, it has one real repeated root (a double root). If Δ < 0, it has no real roots — instead, the roots are complex or imaginary.
如果 Δ > 0，方程有两个不同实数根。如果 Δ = 0，方程有一个实数重根。如果 Δ < 0，方程没有实数根——此时根为复数或虚数。
In the IGCSE syllabus, you are often asked to determine the number of roots or to find the value of k that gives equal roots. For example, for x² + kx + 9 = 0 to have a double root, the discriminant must be zero: k² – 36 = 0, so k = ±6.
在 IGCSE 考纲中，你经常需要判断根的个数，或求出使方程有相等根的 k 值。例如，要使 x² + kx + 9 = 0 有重根，判别式必须为零：k² – 36 = 0，所以 k = ±6。
7. Sum and Product of Roots | 根的和与积
For a quadratic equation ax² + bx + c = 0 with roots α and β, there are important relationships:
对于根为 α 和 β 的二次方程 ax² + bx + c = 0，有以下重要关系：
α + β = -b/a 且 αβ = c/a
These relationships allow you to construct a quadratic equation if the roots are given. For instance, if the roots are 2 and -3, then the sum is -1 and the product is -6. The equation is x² – (-1)x + (-6) = 0, which is x² + x – 6 = 0.
这些关系允许你在已知根的情况下构造二次方程。例如，如果根是 2 和 -3，那么和为 -1，积为 -6。方程为 x² – (-1)x + (-6) = 0，即 x² + x – 6 = 0。
8. Solving Quadratic Inequalities | 解二次不等式
Quadratic inequalities are closely connected to the graph of the quadratic function. To solve x² – 5x + 6 > 0, first factorise: (x – 2)(x – 3) > 0. The roots are 2 and 3.
二次不等式与二次函数图像密切相关。解 x² – 5x + 6 > 0，先因式分解：(x – 2)(x – 3) > 0。根为 2 和 3。
Sketch the parabola or test intervals. The expression is positive when x < 2 or x > 3, because both factors are either both negative or both positive. Therefore, the solution set is x < 2 or x > 3.
画出抛物线草图或使用区间测试。当 x < 2 或 x > 3 时，表达式为正，因为此时两个因子同号。因此解集为 x < 2 或 x > 3。
For the inequality x² – 5x + 6 < 0, the expression is negative between the roots, so the solution is 2 < x < 3. Remember to reverse the inequality sign if you multiply or divide by a negative number.
对于不等式 x² – 5x + 6 < 0，表达式在两根之间为负，所以解为 2 < x < 3。记住：如果乘以或除以负数，要改变不等号方向。
9. Writing a Quadratic Equation from Its Roots | 由根构造二次方程
Given roots p and q, a quadratic with exactly these roots can be written as (x – p)(x – q) = 0. Expanding this gives x² – (p + q)x + pq = 0. This is a fast way to reconstruct an equation.
已知根 p 和 q，以它们为根的二次方程可写成 (x – p)(x – q) = 0。展开得 x² – (p + q)x + pq = 0。这是快速重现方程的方法。
If the roots are given in surd form, such as 2 + √3 and 2 – √3, then the sum is 4 and the product is (2 + √3)(2 – √3) = 4 – 3 = 1. The equation is x² – 4x + 1 = 0.
如果根以根式形式给出，例如 2 + √3 和 2 – √3，则和为 4，积为 (2 + √3)(2 – √3) = 4 – 3 = 1。方程为 x² – 4x + 1 = 0。
10. Graphical Interpretation | 图像的解读
The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upward; if a < 0, it opens downward. The x-intercepts of the graph correspond to the real roots of the quadratic equation.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，开口向上；如果 a < 0，开口向下。图像与 x 轴的交点对应二次方程的实数根。
The vertex of the parabola can be found using the completed square form. For y = a(x + p)² + q, the vertex is at (-p, q). The axis of symmetry is the vertical line passing through the vertex.
抛物线的顶点可以通过配方法找到。对于 y = a(x + p)² + q，顶点在 (-p, q)。对称轴是过顶点的竖直直线。
A quadratic equation with a positive discriminant will intersect the x-axis at two points. With a zero discriminant, the graph touches the x-axis at exactly one point. With a negative discriminant, the graph never touches the x-axis.
判别式为正的二次方程会与 x 轴有两个交点。判别式为零时，图像与 x 轴相切于一点。判别式为负时，图像与 x 轴无交点。
11. Common Mistakes to Avoid | 常见错误警示
One frequent error is forgetting that if a ≠ 1, the quadratic formula still requires careful substitution. Students also often miscopy signs — for example, treating x² – 4x – 5 = 0 as x² + 4x – 5 = 0. This changes both roots completely.
一个常见错误是当 a ≠ 1 时，忘记求根公式仍需细心代入。学生还常抄错符号——例如把 x² – 4x – 5 = 0 当作 x² + 4x – 5 = 0。这会完全改变两个根。
Another pitfall is not rearranging the equation to standard form before solving. For example, x² = 9 – 6x must first be rewritten as x² + 6x – 9 = 0 before using any method.
另一个常见陷阱是在求解前没有将方程整理为标准形式。例如，x² = 9 – 6x 必须先重写为 x² + 6x – 9 = 0，才能使用任何方法。
Finally, when using the quadratic formula, be careful with the denominator. The entire expression -b ± √… is divided by 2a, not just the square root. Writing x = -b ± (√… / 2a) is incorrect and will lead to wrong answers.
最后，使用求根公式时要注意分母。整个表达式 -b ± √… 都要除以 2a，而不仅仅是被开方数。写成 x = -b ± (√… / 2a) 是错误的，会导致答案错误。
12. Practice Makes Perfect | 熟能生巧
To truly master quadratics, practice a variety of question types. Start by factorising simple cases, then progress to completing the square, then apply the formula to more complex equations. Use past papers to familiarise yourself with the style and timing.
要真正掌握一元二次方程，需要练习不同类型的题目。从简单情形的因式分解开始，然后进阶配方，再将求根公式应用于更复杂的方程。使用历年真题来熟悉题型和时间分配。
Remember that solving quadratic equations is not just an isolated skill — it appears in coordinate geometry, calculus basics, and problem-solving questions throughout the IGCSE syllabus. A deep understanding here will pay dividends across the entire examination.
请记住，解一元二次方程不仅仅是一个孤立知识点——它贯穿于 IGCSE 考纲中的坐标几何、微积分基础及应用题中。在此处的深刻理解将在整个考试中带来丰厚回报。
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