Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most fundamental topics in IGCSE Mathematics. Mastering them opens the door to understanding parabolas, projectile motion, and countless real-world optimisation problems. This guide covers every method you need, from factorisation to the quadratic formula, with exam-style tips throughout.

二次方程是 IGCSE 数学中最基础也最重要的主题之一。掌握它不仅能帮助你理解抛物线、抛体运动，还能解决无数现实中的优化问题。本指南涵盖你所需的所有解法——从因式分解到求根公式，并穿插考试风格的技巧提示。

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1. The Standard Form | 标准形式

Every quadratic equation can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The coefficient a determines whether the parabola opens upward (a > 0) or downward (a < 0).

每一个二次方程都可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。系数 a 决定抛物线的开口方向：当 a > 0 时开口向上，当 a < 0 时开口向下。

ax² + bx + c = 0

For example, 2x² − 5x + 3 = 0 is in standard form with a = 2, b = −5, c = 3. A quadratic equation always has at most two solutions, which are also called roots or solutions.

例如，2x² − 5x + 3 = 0 就是标准形式，其中 a = 2，b = −5，c = 3。二次方程最多有两个解，这些解也称为根或解。

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2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the equation has simple integer roots. The principle is based on the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

当方程具有简单的整数根时，因式分解通常是最快捷的方法。其原理基于零乘积性质：如果两个表达式的乘积为零，那么至少其中一个必须为零。

If A × B = 0, then A = 0 or B = 0

Worked Example: Solve x² − 7x + 12 = 0.

例题：解方程 x² − 7x + 12 = 0。

• Find two numbers that multiply to +12 and add to −7. These are −3 and −4.
• 找到两个数：乘积为 +12，和为 −7。这两个数是 −3 和 −4。
• Factorise: (x − 3)(x − 4) = 0.
• 因式分解：(x − 3)(x − 4) = 0。
• Apply zero product property: x − 3 = 0 or x − 4 = 0.
• 应用零乘积性质：x − 3 = 0 或 x − 4 = 0。
• Solutions: x = 3 or x = 4.
• 解为：x = 3 或 x = 4。

When the coefficient of x² is not 1, you may need to factorise by grouping. For example, 2x² + 7x + 3 = 0 requires finding two numbers whose product is 2 × 3 = 6 and whose sum is 7 — those numbers are 1 and 6, allowing us to rewrite and group.

当 x² 的系数不为 1 时，你可能需要使用分组分解法。例如，2x² + 7x + 3 = 0 需要找到两个数，其乘积为 2 × 3 = 6，和为 7——这两个数是 1 和 6，从而可以重新书写并分组。

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3. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic as a perfect square plus a constant. This method is especially useful when factorisation is not possible and also helps in finding the vertex of a parabola.

配方法将二次方程改写为一个完全平方加上一个常数的形式。当因式分解不可行时，此方法尤为有用，同时也有助于求抛物线的顶点。

Worked Example: Solve x² + 6x − 7 = 0 by completing the square.

例题：用配方法解 x² + 6x − 7 = 0。

• Move the constant to the right: x² + 6x = 7.
• 将常数移到右边：x² + 6x = 7。
• Add the square of half the coefficient of x: (6/2)² = 9 to both sides.
• 两边同时加上 x 系数一半的平方：(6/2)² = 9。
• x² + 6x + 9 = 16, which gives (x + 3)² = 16.
• x² + 6x + 9 = 16，即 (x + 3)² = 16。
• Take square roots: x + 3 = ±4.
• 两边开平方：x + 3 = ±4。
• Hence x = 1 or x = −7.
• 因此 x = 1 或 x = −7。

For the general quadratic ax² + bx + c = 0 with a = 1, the completed square form is (x + b/2)² − (b/2)² + c = 0.

对于一般二次方程 ax² + bx + c = 0（当 a = 1 时），配方形式为 (x + b/2)² − (b/2)² + c = 0。

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4. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation, regardless of whether the roots are rational, irrational, or even complex. It is derived from completing the square and is your most reliable tool.

求根公式适用于所有二次方程，无论根是有理数、无理数还是复数。它由配方法推导而来，是你最可靠的工具。

x = (−b ± √(b² − 4ac)) / 2a

Worked Example: Solve 3x² − 5x − 2 = 0 using the formula.

例题：用求根公式解 3x² − 5x − 2 = 0。

• Identify a = 3, b = −5, c = −2.
• 确定 a = 3，b = −5，c = −2。
• Calculate the discriminant: b² − 4ac = 25 + 24 = 49.
• 计算判别式：b² − 4ac = 25 + 24 = 49。
• Substitute into the formula: x = (5 ± √49) / 6 = (5 ± 7) / 6.
• 代入公式：x = (5 ± √49) / 6 = (5 ± 7) / 6。
• Solutions: x = 2 or x = −1/3.
• 解为：x = 2 或 x = −1/3。

You should memorise the formula and practice using it with negative coefficients carefully, as sign errors are the most common mistake.

你应该牢记这个公式，并注意带负系数的代入练习，因为符号错误是最常见的失误。

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5. The Discriminant | 判别式

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

判别式，记作 Δ = b² − 4ac，无需完全解方程即可判断二次方程有多少个实数根。

Discriminant | 判别式	Number of Real Roots | 实数根个数
Δ > 0 (perfect square)	2 distinct rational roots | 两个不同的有理根
Δ > 0 (not a perfect square)	2 distinct irrational roots | 两个不同的无理根
Δ = 0	1 repeated root | 一个重根
Δ < 0	No real roots | 无实数根

For example, the equation x² + 2x + 5 = 0 has Δ = 4 − 20 = −16 < 0, so it has no real solutions — its graph does not intersect the x-axis.

例如，方程 x² + 2x + 5 = 0 的判别式 Δ = 4 − 20 = −16 < 0，因此没有实数解——其图像不与 x 轴相交。

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6. Solving by the Graph Method | 图像法

Graphically, the solutions of a quadratic equation are the x-coordinates where the parabola y = ax² + bx + c crosses the x-axis. These intersection points correspond to y = 0.

从图像上看，二次方程的解就是抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。这些交点对应于 y = 0。

When asked to solve graphically, you draw the parabola accurately using a table of values, then read off the x-intercepts. The answers obtained graphically are approximate unless the roots are integers.

题目要求用图像法求解时，你需要通过列表取值准确画出抛物线，然后读出与 x 轴的交点。除非根是整数，否则图像法得到的答案通常是近似值。

For the equation x² − 2x − 3 = 0, the graph crosses the x-axis at x = −1 and x = 3, which can be verified by factorisation: (x − 3)(x + 1) = 0.

对于方程 x² − 2x − 3 = 0，图像与 x 轴交于 x = −1 和 x = 3，这可以通过因式分解验证：(x − 3)(x + 1) = 0。

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7. Solving with a GDC | 图形计算器求解

In the IGCSE exam, you may be allowed to use a graphing calculator (GDC). Using the polynomial solver or graphing function can give you exact or approximate roots quickly, but you should still know the algebra to verify your answers.

在 IGCSE 考试中，你可能会被允许使用图形计算器 (GDC)。使用多项式求解功能或绘图功能可以快速得到精确或近似根，但你仍应掌握代数方法以验证答案。

• Enter the equation into the graphing screen and identify the zeros.
• 在绘图界面输入方程并找出零点。
• Use the “poly” or “solve” function directly for exact values where possible.
• 尽可能直接使用“poly”或“solve”功能获得精确值。
• Always round to 3 significant figures unless the question specifies otherwise.
• 除非题目另有要求，否则始终保留 3 位有效数字。

Remember: showing your working in the exam earns method marks, even if the final answer comes from a calculator.

请记住：在考试中展示解题过程可以获得方法分，即使最终答案来自计算器。

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8. Quadratic Equations from Word Problems | 应用题中的二次方程

Many exam questions present real-life scenarios that lead to quadratic equations. The key steps are: define a variable, translate the information into an equation, solve it, and interpret the solution in context.

许多考试题目会给出导致二次方程的现实场景。关键步骤是：设定变量、将信息转化为方程、解方程、并在情境中解释解的含义。

Worked Example: A rectangle has length 5 cm longer than its width, and its area is 84 cm². Find the dimensions.

例题：一个长方形的长比宽多 5 cm，面积为 84 cm²。求其尺寸。

• Let width = w, then length = w + 5.
• 设宽为 w，则长为 w + 5。
• Area: w(w + 5) = 84.
• 面积：w(w + 5) = 84。
• This gives w² + 5w − 84 = 0.
• 得到 w² + 5w − 84 = 0。
• Factorise: (w + 12)(w − 7) = 0, so w = 7 or w = −12.
• 因式分解：(w + 12)(w − 7) = 0，所以 w = 7 或 w = −12。
• Since width cannot be negative, w = 7 cm and length = 12 cm.
• 由于宽度不能为负，w = 7 cm，长度 = 12 cm。

Always reject solutions that do not make sense in the context, such as negative lengths or times.

始终淘汰在情境中不合理的解，如负的长度或时间。

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9. 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 relationships hold, derived directly from the factorised form:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，以下关系直接由因式分解形式推导得出：

Sum of roots: α + β = −b/a

Product of roots: α × β = c/a

These relationships allow you to quickly check your solutions or to construct a quadratic equation when the roots are known. For example, if the roots are 2 and −5, the equation is x² − (sum)x + product = 0, giving x² + 3x − 10 = 0.

这些关系可以让你快速验证解，或在已知根时构造二次方程。例如，若根为 2 和 −5，方程为 x² − (和)x + 积 = 0，即 x² + 3x − 10 = 0。

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10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Avoiding common pitfalls is essential for scoring full marks on quadratic equation questions.

避免常见陷阱对于在二次方程题目中拿到满分至关重要。

Common Mistake | 常见错误	Correct Approach | 正确做法
Forgetting to set the equation to 0 before factorising	Always rearrange to ax² + bx + c = 0 first
Sign errors in the quadratic formula	Write b = −5 explicitly and substitute carefully
Dropping one of the ± roots	Always write both solutions
Not showing method marks	Show factorisation or substitution steps clearly

Remember the order in an exam: 1. Set to 0 → 2. Factorise or formula → 3. Write both roots.

考试中的顺序：1. 化为标准形式 → 2. 因式分解或公式 → 3. 写出两个根。

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11. Practice Questions | 练习题目

Try these problems on your own, then check your answers.

请先独立尝试以下题目，再核对答案。

Question | 题目	Answer | 答案
1. Solve x² − 5x − 14 = 0	x = 7 或 x = −2
2. Solve 2x² + 3x − 5 = 0	x = 1 或 x = −2.5
3. Solve x² + 4x + 1 = 0 (give exact form)	x = −2 ± √3
4. Find k if x² + kx + 9 = 0 has exactly one root	k = ±6

For question 4, set the discriminant Δ = k² − 36 = 0, giving k = ±6 — the parabola touches the x-axis at a single point.

对于第 4 题，令判别式 Δ = k² − 36 = 0，得到 k = ±6——此时抛物线与 x 轴相切于一点。

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12. Summary | 总结

You now have all the tools to tackle quadratic equations with confidence. Factorisation is fastest when roots are simple; completing the square reveals the vertex; the formula never fails; and the discriminant provides a quick reality check. Practice each method until it becomes automatic — this is a skill that pays dividends across algebra, functions, and coordinate geometry.

你现在已经拥有自信应对二次方程的所有工具。根简单时因式分解最快；配方法能揭示顶点；求根公式永远不会失效；判别式则能快速检验根的个数。将每个方法练习到自动化的程度——这项技能在代数、函数和坐标几何中都会带来丰厚的回报。

In your exam, read the question carefully, choose the most efficient method, show all working, and interpret your answers in context if required. Good luck!

考试时，请仔细读题，选择最高效的方法，写出完整的解题过程，并根据题目要求解释答案的实际意义。祝你好运！

Published by TutorHao | Mathematics Revision Series | aleveler.com

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