Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in almost every exam paper, whether as direct solving questions, word problems, or graph-related tasks. This article will guide you through everything you need to know: standard forms, solving methods, the discriminant, and common exam traps.

二次方程是 IGCSE 数学中最重要的主题之一。几乎每份试卷都会出现它的身影，无论是直接解方程、应用题，还是与图像相关的题目。本文将带你掌握所有必备知识：标准形式、求解方法、判别式以及常见考试陷阱。

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1. Standard Form | 标准形式

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 x is 2, which is why it is called “quadratic” (from the Latin “quadratus”, meaning square).

其中 a、b、c 为常数，且 a ≠ 0。未知数 x 的最高次数为 2，因此称为”二次”方程。

• If a = 0, the equation becomes linear, not quadratic.
• 如果 a = 0，方程变为一次方程，而非二次方程。
• The values of x that satisfy the equation are called its roots or solutions.
• 满足方程的 x 值称为方程的根或解。
• A quadratic equation can have at most two distinct real roots.
• 二次方程最多有两个不同的实数根。

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2. Solving by Factorisation | 因式分解法

Factorisation is the most direct method when the equation has simple integer coefficients. The idea is to rewrite ax² + bx + c into the product of two brackets, then set each bracket equal to zero.

当方程具有简单的整数系数时，因式分解是最直接的方法。其核心思想是将 ax² + bx + c 改写为两个括号的乘积，然后令每个括号分别等于零。

For example, solve x² – 5x + 6 = 0:

例如，解方程 x² – 5x + 6 = 0：

(x – 2)(x – 3) = 0

x = 2 or x = 3

• Step 1: Factorise the quadratic expression.
• 步骤一：对二次表达式进行因式分解。
• Step 2: Set each factor equal to zero.
• 步骤二：令每个因式等于零。
• Step 3: Solve the resulting linear equations.
• 步骤三：解所得的一次方程。
• Always expand your brackets to check your factors are correct.
• 务必展开括号以检验因式是否正确。

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3. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula always works. For the equation ax² + bx + c = 0, the roots are given by:

当因式分解困难或无法进行时，求根公式始终适用。对于方程 ax² + bx + c = 0，其根由下式给出：

x = (−b ± √(b² − 4ac)) ⁄ (2a)

This formula is provided in most exams, but you must know how to substitute values correctly and simplify the result.

多数考试会提供这一公式，但你必须掌握正确的代入方法和结果化简技巧。

• Write down the values of a, b and c first to avoid mistakes.
• 先写出 a、b、c 的值，避免出错。
• Calculate the discriminant b² − 4ac before continuing.
• 先计算判别式 b² − 4ac，再进行后续计算。
• Use a calculator for the final arithmetic where allowed.
• 在允许的情况下，用计算器完成最后的算术运算。

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4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is especially useful for finding turning points and solving equations when factorisation is messy.

配方法将二次式改写为 a(x + p)² + q 的形式。这在求顶点坐标以及因式分解不便时求解方程时尤其有用。

For example, for x² + 6x + 5 = 0:

例如，对于 x² + 6x + 5 = 0：

x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4

Then solve:

然后求解：

(x + 3)² − 4 = 0 → (x + 3)² = 4

x + 3 = ±2 → x = −1 or x = −5

• Halve the coefficient of x to find p.
• 取 x 系数的一半可求出 p。
• Subtract the square of p to keep the expression equivalent.
• 减去 p 的平方以保持等式成立。

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5. The Discriminant | 判别式

The discriminant is the expression b² − 4ac under the square root sign. It tells us about the roots without fully solving the equation.

判别式是根号下的表达式 b² − 4ac。它无需完整解方程即可判断根的情况。

b² − 4ac	Nature of Roots | 根的性质
> 0	Two distinct real roots | 两个不同的实数根
= 0	One repeated root | 一个重根
< 0	No real roots | 无实数根

In IGCSE exams, you will often be asked to find the range of k for which a given quadratic has real roots. Write the discriminant condition as an inequality and solve it.

在 IGCSE 考试中，常要求你求出使给定二次方程有实数根时 k 的取值范围。将判别式条件写成不等式并求解即可。

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6. Sum and Product of Roots | 根的和与积

For the quadratic equation ax² + bx + c = 0 with roots α and β, we have two important relationships:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，有两个重要的关系式：

Sum of roots: α + β = −b ⁄ a

Product of roots: αβ = c ⁄ a

These relationships allow you to form a new quadratic equation from given roots, or to find values of expressions like α² + β² without solving for the roots.

利用这些关系，可以从已知的根构造新的二次方程，或者在不解出根的情况下求出 α² + β² 等表达式的值。

• Note the negative sign in the sum formula.
• 注意求和公式中的负号。
• For a monic quadratic x² + bx + c = 0, the sum is −b and the product is c.
• 对于首项系数为 1 的二次方程 x² + bx + c = 0，根和为 −b，根积为 c。

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7. Quadratic Graphs | 二次函数图像

The graph of y = ax² + bx + c is a parabola. Its shape depends on the sign of a.

y = ax² + bx + c 的图像是一条抛物线，其形状取决于 a 的符号。

• If a > 0: the parabola opens upwards and has a minimum point.
• 若 a > 0：抛物线开口向上，有最小值点。
• If a < 0: the parabola opens downwards and has a maximum point.
• 若 a < 0：抛物线开口向下，有最大值点。
• The x-intercepts are the roots of the equation.
• 图像与 x 轴的交点即为方程的根。
• The y-intercept is c.
• 图像与 y 轴的交点为 c。
• The turning point has x-coordinate −b ⁄ (2a) and y-coordinate found by substitution.
• 顶点的 x 坐标为 −b ⁄ (2a)，y 坐标通过代入求得。

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8. Word Problems | 应用题

Quadratic equations often arise from geometric or numeric word problems. For example, “The length of a rectangle is 3 cm more than its width, and its area is 40 cm². Find the width.”

二次方程常出现在几何或数值应用题中。例如：”一个长方形的长比宽多 3 cm，面积为 40 cm²，求宽。”

Let width = x → length = x + 3 → x(x + 3) = 40

x² + 3x − 40 = 0 → (x + 8)(x − 5) = 0 → x = 5

Note that x = −8 is discarded because a length cannot be negative. In applied problems, always check whether your solutions make sense in context.

注意 x = −8 应舍去，因为长度不能为负数。在应用题中，务必检查解是否符合实际情况。

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9. Common Exam Mistakes | 常见考试失误

Many students lose marks needlessly on quadratic questions. Here are the most frequent pitfalls to avoid:

许多学生在二次方程题目上无谓失分。以下是最常见的陷阱，务必避免：

• Forgetting to rearrange all terms to one side first.
• 忘记先将所有项移到等号一边。
• Making sign errors when substituting into the formula.
• 代入公式时出现符号错误。
• Dropping the ± symbol in the quadratic formula.
• 在求根公式中遗漏 ± 符号。
• Dividing both sides by a variable that could be zero.
• 等式两边同时除以可能为零的未知数。
• Misstating the discriminant condition for “two equal roots” as > 0 instead of = 0.
• 将”两个相等根”的判别式条件误写为 > 0 而非 = 0。

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10. Practice Strategy | 练习策略

To achieve top marks, you must be comfortable using all three solving methods and know when to choose each one.

要取得高分，你必须熟练运用三种求解方法，并知道何时选择哪一种。

Method | 方法	When to Use | 适用时机
Factorisation | 因式分解	Simple integer coefficients | 简单整数系数
Completing the square | 配方法	Finding turning point or dealing with surd roots | 求顶点或处理无理根
Quadratic formula | 求根公式	Any equation, especially non-factorisable ones | 任何方程，尤其是不可因式分解的

Practice at least ten questions of each type. Time yourself, and always write out every step clearly — examiners award method marks even if the final answer is wrong.

每类题目至少练习十道。计时完成，并始终清晰地写出每一步——即使最终答案错误，阅卷官也会给方法分。

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11. Summary | 总结

Quadratic equations are a systematic topic: standard form, solving methods, discriminant, graphs, and applications. Once you master each part, questions become predictable and manageable.

二次方程是一个体系化的主题：标准形式、求解方法、判别式、图像和应用。一旦你掌握了每一个部分，题目就会变得可预测且易于处理。

Remember to practice past papers under timed conditions, and use the discriminant every time you want to check the nature of roots. With consistent effort, this topic will become one of your strongest areas.

切记在限时条件下练习真题，并在每次想判断根的性质时使用判别式。只要坚持不懈，这一主题将成为你最强的得分领域之一。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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