Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, geometry, and even real-world problem solving. Mastering them opens the door to higher-level maths.

一元二次方程是 IGCSE 数学中最重要的内容之一。它们出现在代数、图像、几何甚至现实问题解决中。掌握它们，你就能打开通往更高水平数学的大门。

1. What Is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The highest power of the variable is 2.

一元二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。变量的最高次数是 2。

Example: 2x² + 3x − 5 = 0 | 例如：2x² + 3x − 5 = 0
Not quadratic: x³ + 2x = 0 | 不是二次：x³ + 2x = 0
Not quadratic (unless a = 0): ax + b = 0 | 当最高次为1时不是二次

2. Standard Form and Coefficients | 标准形式与系数

Always rearrange the equation into standard form before solving. In ax² + bx + c = 0, the coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term.

在解题前，务必把方程整理成标准形式。在 ax² + bx + c = 0 中，a 是二次项系数，b 是一次项系数，c 是常数项。

ax² + bx + c = 0, where a ≠ 0

For example, in 3x² − 6x + 2 = 0, we have a = 3, b = −6, c = 2. If an equation like 2x² = 5x − 1 is given, first rewrite it as 2x² − 5x + 1 = 0.

例如，在 3x² − 6x + 2 = 0 中，a = 3，b = −6，c = 2。如果给出 2x² = 5x − 1，先改写为 2x² − 5x + 1 = 0。

3. Solving by Factorisation | 用因式分解法求解

If a quadratic expression can be factorised into two linear factors, the solutions are found by setting each factor equal to zero.

如果二次表达式可以被分解为两个一次因式，那么分别令每个因式等于零，即可求出解。

If pq = 0, then p = 0 or q = 0

Example: Solve x² − 5x + 6 = 0. Factorise: (x − 2)(x − 3) = 0. So x = 2 or x = 3.

例：解 x² − 5x + 6 = 0。因式分解：(x − 2)(x − 3) = 0。所以 x = 2 或 x = 3。

Always check if the coefficient of x² is 1 first | 先检查 x² 的系数是否为 1
If a ≠ 1, use the “multiply a × c” method | 如果 a ≠ 1，使用“a × c”乘积法
Be careful with signs when factorising | 因式分解时注意符号

4. Solving by Completing the Square | 用配方法求解

Completing the square rewrites the equation in the form (x + p)² = q. This is especially useful when the quadratic cannot be factorised easily.

配方法将方程改写为 (x + p)² = q 的形式。当二次式不易因式分解时，这种方法特别有用。

x² + bx = (x + b/2)² − (b/2)²

Example: Solve x² + 6x − 1 = 0. Complete the square: (x + 3)² − 9 − 1 = 0 → (x + 3)² = 10. Then x = −3 ± √10.

例：解 x² + 6x − 1 = 0。配方：(x + 3)² − 9 − 1 = 0 → (x + 3)² = 10。因此 x = −3 ± √10。

Ensure the coefficient of x² is 1 before completing the square | 配方前确保 x² 系数为 1
If a ≠ 1, divide the whole equation by a first | 若 a ≠ 1，先将整个方程除以 a
The method also helps find the turning point of a parabola | 配方法还能帮助找到抛物线的顶点

5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation directly. It is derived from completing the square and works for all values of a, b, c.

求根公式可以直接解出任何一元二次方程。它由配方法推导而来，适用于所有 a、b、c 的取值。

x = (−b ± √(b² − 4ac)) / (2a)

Example: Solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2. Substitute: x = [−3 ± √(9 + 16)] / 4 = [−3 ± 5] / 4. So x = 0.5 or x = −2.

例：解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。代入：x = [−3 ± √(9 + 16)] / 4 = [−3 ± 5] / 4。所以 x = 0.5 或 x = −2。

Be careful with negative signs when substituting b | 代入 b 时注意负号
Simplify √(b² − 4ac) first | 先化简 √(b² − 4ac)
The formula is given in the formula sheet in many exams | 许多考试中公式表会给出求根公式

6. The Discriminant and Nature of Roots | 判别式与根的性质

The discriminant, denoted by Δ = b² − 4ac, tells us how many real roots a quadratic equation has without solving it.

判别式，记作 Δ = b² − 4ac，无需解方程就能告诉我们该二次方程有几个实数根。

Discriminant Δ	Nature of Roots	根的性质
Δ > 0	Two distinct real roots	两个不相等的实数根
Δ = 0	One repeated real root (equal roots)	一个重根（两个相等实数根）
Δ < 0	No real roots	无实数根

Example: For 3x² − x + 2 = 0, Δ = (−1)² − 4(3)(2) = 1 − 24 = −23, so there are no real roots.

例：对于 3x² − x + 2 = 0，Δ = (−1)² − 4(3)(2) = 1 − 24 = −23，所以没有实数根。

7. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the following relations hold:

对于二次方程 ax² + bx + c = 0，若其两根为 α 和 β，则有以下关系：

α + β = −b/a, αβ = c/a

This is useful when you need to construct a quadratic equation from given roots, or check the correctness of your solutions.

当你需要根据已知根构造二次方程，或检查解是否正确时，这些关系非常有用。

If α + β = 5 and αβ = 6, the equation is x² − 5x + 6 = 0 | 若 α + β = 5，αβ = 6，则方程为 x² − 5x + 6 = 0
Always divide the original equation by a if necessary | 必要时先将原方程除以 a
These formulas also apply to complex roots | 这些公式也适用于复数根

8. Word Problems with Quadratic Equations | 二次方程应用题

Many real-life problems involve quadratic relationships, such as areas, projectile motion, and number puzzles.

许多实际问题涉及二次关系，如面积、抛体运动、数字谜题等。

Example: The length of a rectangle is 3 cm longer than its width, and the area is 28 cm². Find the width.

例：一个长方形的长比宽多 3 cm，面积为 28 cm²。求宽。

Let width = x. Then length = x + 3. So x(x + 3) = 28 → x² + 3x − 28 = 0. Factorise: (x + 7)(x − 4) = 0. Since width cannot be negative, x = 4 cm.

设宽为 x，则长为 x + 3。所以 x(x + 3) = 28 → x² + 3x − 28 = 0。因式分解：(x + 7)(x − 4) = 0。由于宽不能为负，所以 x = 4 cm。

Define a variable clearly | 清晰地定义变量
Translate the problem into a quadratic equation | 将问题转化为二次方程
Reject solutions that do not make sense in context | 舍去不符合实际意义的解

9. Quadratic Graphs and Intersections | 二次函数图像与交点

The solutions of a quadratic equation ax² + bx + c = 0 correspond to the x-coordinates where the parabola y = ax² + bx + c crosses the x-axis.

二次方程 ax² + bx + c = 0 的解，对应抛物线 y = ax² + bx + c 与 x 轴交点的 x 坐标。

If Δ > 0, the parabola crosses the x-axis twice | 若 Δ > 0，抛物线两次穿过 x 轴
If Δ = 0, the parabola touches the x-axis at its vertex | 若 Δ = 0，抛物线在顶点处与 x 轴相切
If Δ < 0, the parabola does not touch the x-axis at all | 若 Δ < 0，抛物线完全不与 x 轴相交
The vertex can be found using x = −b/(2a) | 顶点横坐标可用 x = −b/(2a) 求得

Graphically solving a quadratic often involves sketching or using a calculator to locate the roots.

图像法解二次方程通常涉及画草图或使用计算器定位根。

10. Common Mistakes and How to Avoid Them | 常见错误与避坑指南

Students often lose marks due to small but critical errors. Knowing these traps helps you stay accurate.

学生常常因细小但关键的失误而丢分。了解这些陷阱能帮助你保持准确。

Mistake	Correct Approach	错误	正确做法
Forgetting to set = 0	Always rearrange to standard form first	忘记将方程整理为 = 0	始终先整理成标准形式
Dropping negative signs	Rewrite b and c carefully before substitution	丢失负号	代入前仔细确认 b 和 c 的符号
Cancelling variable factors incorrectly	Never divide both sides by x if x could be zero	错误约去含变量的因式	若 x 可能为 0，绝不同时除以 x
Not checking answers	Substitute roots back into the original equation	不检验答案	将根代回原方程验证

11. Choosing the Best Method | 如何选择最佳方法

Different quadratics suit different methods. Making the right choice saves time in exams.

不同类型的二次方程适合不同的方法。在考试中做出正确选择能节省时间。

Factorisation: Use when the roots are integers or simple rationals | 因式分解法：当根为整数或简单有理数时使用
Completing the square: Use when the coefficient of x² is 1 and b is even | 配方法：当 x² 系数为 1 且 b 为偶数时使用
Quadratic formula: Use for any equation, especially when roots are irrational or complicated | 求根公式：适用于任何方程，尤其当根为无理数或较复杂时
Graphing: Use to estimate roots or check answers | 图像法：用于估算根或检查答案

Always read the question carefully. If it asks for exact values, use algebra rather than a decimal approximation from a graph.

务必仔细审题。如果要求精确值，应使用代数方法而不是从图像中获取近似小数。

12. Practice Questions | 练习题

Try these problems by yourself before checking the solutions.

请先自己尝试以下题目，再对照答案。

Solve x² − 7x + 10 = 0 | 解方程 x² − 7x + 10 = 0
Solve 2x² + 5x − 3 = 0 using the quadratic formula | 用求根公式解 2x² + 5x − 3 = 0
Find the value of k if x² + kx + 16 = 0 has exactly one real root | 若 x² + kx + 16 = 0 恰有一个实数根，求 k 的值
Write a quadratic equation whose roots are 3 and −2 | 写出一个两根为 3 和 −2 的二次方程

Answers: (1) x = 2 or x = 5; (2) x = 0.5 or x = −3; (3) k = ±8; (4) x² − x − 6 = 0

答案：(1) x = 2 或 x = 5；(2) x = 0.5 或 x = −3；(3) k = ±8；(4) x² − x − 6 = 0

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Solving Quadratic Equations | 解一元二次方程