📚 Solving Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程求解全攻略
Quadratic equations are a central topic in the IGCSE Mathematics syllabus. They appear in algebra, graphs, and problem-solving questions, and candidates are expected to solve them by factorising, using the quadratic formula, and completing the square. A strong understanding of quadratics also supports work on inequalities, simultaneous equations, and functions. This article reviews the key methods, common pitfalls, and exam strategies you need to master this topic with confidence.
二次方程是 IGCSE 数学课程的核心主题之一。它出现在代数、图像和问题解决类题目中，考生需要掌握因式分解法、二次公式法和配方法等求解方法。扎实理解二次方程还有助于学习不等式、联立方程和函数等内容。本文梳理关键方法、常见错误和考试策略，帮助你有信心地攻克这一主题。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
nts and a cannot be zero. If a were zero, the equation would become linear rather than quadratic.
任何可以整理为 ax² + bx + c = 0 形式的方程都是二次方程，其中 a、b、c 为常数且 a 不能为 0。如果 a 为 0，方程就变成一次方程，而不是二次方程。
c is the constant term. For example, in 3x² − 5x + 2 = 0, we have a = 3, b = −5 and c = 2.
系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。例如，在 3x² − 5x + 2 = 0 中，a = 3，b = −5，c = 2。
Before solving any quadratic equation, it is essential to write it in standard form. This makes the values of a, b and c clear and helps avoid sign mistakes later.
在解任何二次方程之前，先将其写成标准形式非常重要。这样能让 a、b、c 的值清晰明确，也有助于避免后续的符号错误。
2. Solving by Factorising | 因式分解法
= 0, then px + q = 0 or rx + s = 0. Setting each factor equal to zero gives the solutions.
当二次式可以写成两个一次因式的乘积时，可以利用零乘积性质：如果 (px + q)(rx + s) = 0，则 px + q = 0 或 rx + s = 0。令每个因式等于 0，即可得到方程的解。
o x = 2 or x = 3. Always check by substituting the values back into the original equation.
例题：解 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。始终要把解代入原方程进行检验。
rs that multiply to give ac and add to give b when a = 1, or use grouping when a is not 1.
当根为整数或简单分数时，因式分解法通常是最快的方法。当 a = 1 时，寻找两个数，使它们的乘积等于 ac，和等于 b；当 a 不等于 1 时，可以使用分组分解法。
3. Using the Quadratic Formula | 二次公式法
The quadratic formula solves any quadratic equation ax² + bx + c = 0:
二次公式可以求解任何形如 ax² + bx + c = 0 的二次方程：
x = (−b ± √b² − 4ac) ÷ 2a
This method is especially useful when the quadratic does not factorise easily.
当二次式不易因式分解时，这个方法尤其有用。
2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2. Substitute into the formula to get x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4, giving x = 0.5 or x = −2.
例题：解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。代入公式得 x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4，所以 x = 0.5 或 x = −2。
You should memorise the quadratic formula for the exam. It is given on some formula sheets, but writing it from memory saves time and reduces the chance of copying errors.
你应该在考前熟记二次公式。虽然有些试卷会提供公式表，但凭记忆写出公式能节省时间，并减少抄写错误。
4. Completing the Square | 配方法
Completing the square transforms x² + bx into (x + b/2)² − (b/2)². For a general quadratic ax² + bx + c, we first factor out a from the x terms, then complete the square inside the bracket.
配方法将 x² + bx 变形为 (x + b/2)² − (b/2)²。对于一般二次式 ax² + bx + c，首先把 x 的项提出系数 a，再在括号内完成配方。
x = (x + 3)² − 9, so the equation becomes (x + 3)² − 5 = 0. Thus (x + 3)² = 5 and x = −3 ± √5.
例题：用配方法解 x² + 6x + 4 = 0。先把 x² + 6x 写成 (x + 3)² − 9，方程变为 (x + 3)² − 5 = 0。因此 (x + 3)² = 5，x = −3 ± √5。
Completing the square is also the method used to derive the quadratic formula and to find the vertex of a parabola, so it is worth practising even if you prefer other solving methods.
配方法也是推导二次公式和求抛物线顶点所用的方法，因此即使你更喜欢其他解法，也值得多加练习。
5. Discriminant and Nature of Roots | 判别式与根的性质
Δ = b² − 4ac. It tells us the nature of the roots without solving the equation completely.
二次方程 ax² + bx + c = 0 的判别式为 Δ = b² − 4ac。它可以在不完全解方程的情况下告诉我们根的性质。
real root. If Δ < 0, there are no real roots, but two complex roots at advanced level.
如果 Δ > 0，方程有两个不同的实根；如果 Δ = 0，方程有一个重根；如果 Δ < 0，方程没有实根（在更高阶学习中会有两个复数根）。
the discriminant is b² − 4ac = 16 − 16 = 0, so the equation has one repeated root, x = 2.
例题：对于 x² − 4x + 4 = 0，判别式为 b² − 4ac = 16 − 16 = 0，因此方程有一个重根 x = 2。
6. Quadratic Graphs and Roots | 二次图像与根
d. The roots of ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.
y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上；若 a < 0，开口向下。方程 ax² + bx + c = 0 的根就是图像与 x 轴交点的横坐标。
mpleting the square or using x = −b ÷ 2a. This is useful for maximum and minimum problems.
抛物线的顶点可以通过配方法或使用 x = −b ÷ 2a 求出。这在最大值和最小值问题中非常有用。
If the discriminant is positive, the parabola crosses the x-axis twice; if it is zero, the parabola touches the x-axis at one point; if it is negative, the parabola does not meet the x-axis at all.
如果判别式为正，抛物线与 x 轴相交两次；如果判别式为 0，抛物线在一点处与 x 轴相切；如果判别式为负，抛物线与 x 轴完全没有交点。
7. Solving Quadratic Inequalities | 解二次不等式
+ 3 = 0 to find the critical values x = 1 and x = 3. Then test intervals on a number line.
解诸如 x² − 4x + 3 > 0 的二次不等式时，先解对应的方程 x² − 4x + 3 = 0，得到临界值 x = 1 和 x = 3，然后在数轴上检验各区间。
For a positive leading coefficient, the quadratic expression is positive outside the critical values and negative between them. Therefore x² − 4x + 3 > 0 gives x < 1 or x > 3.
当二次项系数为正时，二次式在临界值之外为正，在两者之间为负。因此 x² − 4x + 3 > 0 的解为 x < 1 或 x > 3。
Always sketch a quick graph or number line when solving quadratic inequalities. This visual check helps you decide whether to take the inside or outside interval.
解二次不等式时，始终快速画出图像或数轴。这种直观检查能帮助你判断应取两个临界值之间的区间还是两侧的区间。
8. Applications and Word Problems | 应用题与文字题
Many IGCSE problems ask you to form a quadratic equation from a real-life situation, such as area, product of numbers, or projectile motion. Identify the unknown, write an equation in terms of x, then solve and check whether answers are sensible.
许多 IGCSE 题目要求根据实际情境建立二次方程，例如面积、数字乘积或抛体运动。先设未知数，用 x 写出方程，然后求解并检查答案是否合理。
its width, and its area is 40 m². Let width be x, then x(x + 3) = 40, giving x² + 3x − 40
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