📚 Quadratic Equations for IGCSE Mathematics | IGCSE数学二次方程
This article provides a comprehensive guide to quadratic equations for IGCSE Mathematics, designed for teachers preparing students for the extended curriculum. It covers key methods of solving quadratic equations, the discriminant, and practical applications, with a strong focus on examination-style accuracy.
本文面向准备 IGCSE 数学（扩展课程）的教师，系统讲解二次方程的核心内容：各类求解方法、判别式与实际应用，并紧扣考试要点，确保学术严谨。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The highest power of the variable is 2.
二次方程是指形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量的最高次数为 2。
Every quadratic equation has at most two solutions (roots). These roots may be real and distinct, real and equal, or not real depending on the value of the discriminant.
每个二次方程至多有两个解（根）。根据判别式的值，这两个根可能是两个相等实数、两个不等实数，也可能不是实数。
2. Solving by Factorisation | 因式分解法
When the quadratic expression can be factorised into two linear factors, we use the zero product property: if p × q = 0, then either p = 0 or q = 0.
当二次表达式可以分解为两个一次因式时，我们利用零积性质：若 p × q = 0，则 p = 0 或 q = 0。
Example: Solve x² − 5x + 6 = 0. Factorising gives (x − 2)(x − 3) = 0, so x = 2 or x = 3.
例：解方程 x² − 5x + 6 = 0。因式分解得到 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。
Always rearrange the equation to the standard form before factorising. If the coefficient of x² is not 1, look for factor pairs that multiply to ac and add to b.
因式分解前务必先将方程整理为标准形式。若 x² 的系数不为 1，则需要寻找两个数，使其乘积等于 ac，且和为 b。
3. Solving by Completing the Square | 配方法
Completing the square rewrites the quadratic equation in the form a(x + h)² + k = 0. This method is particularly useful when factorisation is not straightforward.
配方法将二次方程改写成 a(x + h)² + k = 0 的形式。当因式分解不易时，此方法尤为有效。
To complete the square for x² + px + q = 0, use (x + p/2)² − (p/2)² + q = 0.
对于 x² + px + q = 0，配方时使用 (x + p/2)² − (p/2)² + q = 0。
Example: x² + 6x + 2 = 0 → (x + 3)² − 9 + 2 = 0 → (x + 3)² = 7 → x = −3 ± √7
例：x² + 6x + 2 = 0 → (x + 3)² − 9 + 2 = 0 → (x + 3)² = 7 → x = −3 ± √7
When the leading coefficient a is not 1, first divide the entire equation by a, then complete the square.
当首项系数 a 不为 1 时，先将方程两边同除以 a，再进行配方。
4. The Quadratic Formula | 二次求根公式
For any quadratic equation ax² + bx + c = 0, the solutions are given by
对于任意二次方程 ax² + bx + c = 0，其解由下式给出
x = (−b ± √b² − 4ac) / (2a)
x = (−b ± √b² − 4ac) / (2a)
This formula works for every quadratic equation, including those that cannot be factorised. Ensure the equation is in standard form before substituting a, b, and c.
该公式适用于所有二次方程，包括不能因式分解的方程。代入 a、b、c 之前，必须确保方程已经化为标准形式。
For example, in 2x² − 7x + 3 = 0, a = 2, b = −7, c = 3. Substituting gives x = (7 ± √49 − 24) / 4 = (7 ± 5) / 4, so x = 3 or x = 0.5.
例如，在 2x² − 7x + 3 = 0 中，a = 2，b = −7，c = 3。代入得 x = (7 ± √49 − 24) / 4 = (7 ± 5) / 4，因此 x = 3 或 x = 0.5。
5. The Discriminant | 判别式
The expression Δ = b² − 4ac is called the discriminant. It determines the nature of the roots without solving the equation.
表达式 Δ = b² − 4ac 称为判别式。它可以在不求根的情况下判断根的性质。
If Δ > 0, the equation has two distinct real roots.
若 Δ > 0，方程有两个不等实数根。
If Δ = 0, the equation has exactly one real root (a repeated root).
若 Δ = 0，方程有一个实数根（重根）。
If Δ < 0, the equation has no real roots.
若 Δ < 0，方程没有实数根。
or example, if kx² + 2x + 1 = 0 has two real roots, then Δ ≥ 0 gives 4 − 4k ≥ 0, so k ≤ 1.
判别式也常用于确定未知系数的取值范围。例如，若 kx² + 2x + 1 = 0 有两个实数根，则 Δ ≥ 0，即 4 − 4k ≥ 0，所以 k ≤ 1。
6. Solving Quadratic Inequalities | 二次不等式
Quadratic inequalities often appear in the extended IGCSE syllabus. To solve them, first solve the corresponding equation, then sketch the parabola to decide the interval.
二次不等式常出现在 IGCSE 扩展教学大纲中。求解时，先解对应的方程，再画抛物线草图确定区间。
Example: Solve x² − 3x − 4 > 0 → (x − 4)(x + 1) > 0 → x < −1 or x > 4
例：解 x² − 3x − 4 > 0 → (x − 4)(x + 1) > 0 → x < −1 或 x > 4
For a quadratic with a > 0, the curve opens upward. The values between the roots give a negative result, and the values outside the roots give a positive result.
当 a > 0 时，抛物线开口向上。两根之间的变量值使表达式取负，两根之外取正。
7. Simultaneous Equations with Quadratics | 含二次方程的联立方程组
IGCSE extended problems often ask you to solve a linear and a quadratic equation simultaneously. Use substitution to eliminate one variable.
IGCSE 扩展问题常要求联立一个一次方程和一个二次方程。使用代入法消去一个变量。
Example: Substitute to get x² − 1 = 2x + 1 → x² − 2x − 2 = 0. Using the quadratic formula, x = 1 ± √3, then find the corresponding y values.
例：联立解 y = 2x + 1 与 y = x² − 1。代入得 x² − 1 = 2x + 1，即 x² − 2x − 2 = 0。用求根公式得 x = 1 ± √3，再求对应的 y 值。
Always check whether the quadratic factorises; if not, apply the formula. The solutions correspond to the intersection points of the line and the curve.
始终先尝试因式分解；若不能分解，则使用公式法。所得解对应直线与曲线的交点。
8. Problem Solving: Quadratic Word Problems | 二次方程应用问题
Many real-world problems can be modelled by quadratic equations. The key steps are: define a variable, form the equation, solve it, and then check whether the answer is sensible in the context.
许多实际问题可以用二次方程建模。关键步骤是：设变量、列方程、解方程，然后结合实际情况检验答案是否合理。
= 0. Factorising gives (w + 10)(w − 7) = 0, hence w = 7 (since width cannot be negative).
例如，一个长方形的长比宽多 3 cm，面积为 70 cm²。设宽为 w，则 w(w + 3) = 70，即 w² + 3w − 70 = 0。因式分解得 (w + 10)(w − 7) = 0，因此 w = 7（因为宽不能为负）。
9. Sketching Quadratic Graphs | 二次函数图像
+ c helps students see the meaning of roots, the turning point, and the line of symmetry.
理解二次函数 y = ax² + bx + c 的图像，有助于学生掌握根、顶点和对称轴的含义。
The roots are the x-intercepts of the graph.
根就是图像与 x 轴交点的横坐标。
The line of symmetry is x = −b/(2a).
对称轴为 x = −b/(2a)。
The turning point lies on the line of symmetry. If a > 0, it is a minimum; if a < 0, it is a maximum.
顶点位于对称轴上。当 a > 0 时为最小值；当 a < 0 时为最大值。
ates of the turning point directly: y = a(x − h)² + k has minimum point (h, k) when a > 0.
配方法可直接给出顶点坐标：y = a(x − h)² + k 在 a > 0 时，最小值点为 (h, k)。
10. Common Examination Pitfalls | 常见考试易错点
Students often lose marks in quadratic questions due to careless sign errors, forgetting to rearrange to the standard form, or ignoring a negative root in word problems.
学生在二次方程题目中常常因符号错误、忘记化为标准形式，或在应用题中忽略负根而失分。
It is vital to check each solution by substituting back into the original equation. When factorising, always expand your factors to verify that the product matches the original quadratic.
务必将每个解代回原方程进行检验。因式分解后，一定要展开因式，验证其乘积是否与原二次表达式一致。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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