Solving Quadratic Equations: Factorisation, Formula, and Graphs | 解二次方程：因式分解、公式法与图像法

📚 Solving Quadratic Equations: Factorisation, Formula, and Graphs | 解二次方程：因式分解、公式法与图像法

Quadratic equations appear frequently in IGCSE Mathematics, combining algebraic manipulation, problem-solving, and graph interpretation. This guide covers the essential methods for solving quadratics, the discriminant, common pitfalls, and exam-style practice.

二次方程在 IGCSE 数学中高频出现，它将代数变形、问题求解和图像解读融为一体。本指南涵盖解二次方程的核心方法、判别式、常见陷阱以及考试风格练习。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is a polynomial equation where the highest power of the variable is 2. It takes the general form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.

二次方程是变量最高次数为 2 的多项式方程，其一般形式为 ax² + bx + c = 0，其中 a、b、c 为实数，且 a ≠ 0。

The condition a ≠ 0 is essential: if a = 0, the equation becomes linear (bx + c = 0), not quadratic. The values a, b, and c are called coefficients: a is the quadratic coefficient, b is the linear coefficient, and c is the constant term.

条件 a ≠ 0 至关重要：若 a = 0，方程退化为一次方程（bx + c = 0），而非二次方程。a、b、c 统称为系数：a 为二次项系数，b 为一次项系数，c 为常数项。

For example, in the equation 2x² – 3x + 1 = 0, we identify a = 2, b = -3, and c = 1. Recognising these coefficients is the first step to choosing the correct solving method.

例如，在方程 2x² – 3x + 1 = 0 中，可识别 a = 2，b = -3，c = 1。准确识别系数是选择正确解法的第一步。

2. Standard Form and Rearranging | 标准形式与方程整理

Before solving any quadratic equation, always rearrange it into the standard form ax² + bx + c = 0. This means moving all terms to one side, leaving zero on the other.

在解任何二次方程之前，务必将其整理为标准形式 ax² + bx + c = 0，即将所有项移到一边，使另一边为零。

Consider the equation x² = 7x – 10. Subtracting 7x and adding 10 to both sides gives x² – 7x + 10 = 0, which is now in standard form with a = 1, b = -7, c = 10.

以方程 x² = 7x – 10 为例：两边同时减去 7x 并加上 10，得到 x² – 7x + 10 = 0，即为标准形式，其中 a = 1，b = -7，c = 10。

Sometimes the equation may contain brackets or fractions. Expand brackets first, then multiply through by the lowest common denominator to eliminate fractions, and finally collect all terms on one side.

有时方程包含括号或分数。应先去括号，再乘以最小公分母以消去分数，最后将所有项移到一边。

3. Solving by Factorisation | 因式分解法求解

Factorisation is the fastest method when the quadratic has simple integer roots. The principle uses the zero-product property: if the product of two expressions is zero, at least one of them must be zero.

当二次方程具有简单整数根时，因式分解是最快速的方法。其原理是零乘积性质：若两个表达式的乘积为零，则至少其中一个必须为零。

If (px + q)(rx + s) = 0, then px + q = 0 or rx + s = 0

To factorise x² + bx + c, find two numbers that multiply to c and add to b. For example, in x² – 5x + 6 = 0, the numbers -2 and -3 multiply to 6 and add to -5, so the factorisation is (x – 2)(x – 3) = 0.

对 x² + bx + c 进行因式分解，需找到两个数，其乘积为 c、和为 b。例如，在 x² – 5x + 6 = 0 中，-2 和 -3 相乘得 6、相加得 -5，因此分解为 (x – 2)(x – 3) = 0。

Applying the zero-product property gives x – 2 = 0 or x – 3 = 0, so the solutions are x = 2 or x = 3. Always check your factors by expanding them back to the original expression.

应用零乘积性质得 x – 2 = 0 或 x – 3 = 0，因此解为 x = 2 或 x = 3。务必通过展开来验算因式分解是否正确。

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

Completing the square rewrites a quadratic in the form (x + p)² + q, which makes solving straightforward. This method works for any quadratic, even when factorisation is not possible.

配方法将二次式改写为 (x + p)² + q 的形式，从而使求解变得直接。该方法适用于任何二次方程，即使无法因式分解也能使用。

For x² + bx, take half of b and square it: (b/2)². Then write x² + bx = (x + b/2)² – (b/2)². Let us apply this to x² + 6x + 5 = 0.

对于 x² + bx，取 b 的一半并平方得到 (b/2)²，于是 x² + bx = (x + b/2)² – (b/2)²。我们以 x² + 6x + 5 = 0 为例。

(x + 3)² – 9 + 5 = 0 ⇒ (x + 3)² = 4

Taking square roots gives x + 3 = ±2, hence x = -3 + 2 = -1 or x = -3 – 2 = -5. The symbol ± is vital: it represents both the positive and negative square roots.

开平方得 x + 3 = ±2，因此 x = -3 + 2 = -1 或 x = -3 – 2 = -5。符号 ± 至关重要：它表示正、负两个平方根。

When the coefficient of x² is not 1, factor it out first. For 2x² + 8x + 3 = 0, write 2(x² + 4x) + 3 = 0, complete the square inside the bracket, then solve.

当 x² 的系数不为 1 时，先将其提出。对于 2x² + 8x + 3 = 0，写成 2(x² + 4x) + 3 = 0，在括号内配方后再求解。

5. The Quadratic Formula | 二次公式法

The quadratic formula is a universal tool that solves any quadratic equation directly. It is derived from completing the square on the general form ax² + bx + c = 0.

二次公式是解任意二次方程的直接通用工具，它通过对一般形式 ax² + bx + c = 0 配方推导而来。

x = (−b ± √(b² − 4ac)) ⁄ (2a)

To use the formula, substitute the values of a, b, and c into the expression. Take the equation 2x² – 4x – 3 = 0: here a = 2, b = -4, c = -3.

使用公式时，将 a、b、c 的值代入表达式。对于方程 2x² – 4x – 3 = 0：这里 a = 2，b = -4，c = -3。

Substituting gives x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4, which simplifies to x = 1 ± (√10)/2. These are exact values; use your calculator only for decimal approximations when asked.

代入得 x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4，化简为 x = 1 ± (√10)/2。这些是精确值；仅在题目要求近似值时，才用计算器求小数。

6. The Discriminant | 判别式

The discriminant is the expression Δ = b² – 4ac found inside the square root of the quadratic formula. It determines the nature of the roots without solving the full equation.

判别式是二次公式根号内的表达式 Δ = b² – 4ac。它无需完整解方程即可判断根的性质。

If Δ > 0, the equation has two distinct real roots. Geometrically, the parabola crosses the x-axis at two points.

若 Δ > 0，方程有两个不同的实数根。几何上，抛物线在两点处与 x 轴相交。
If Δ = 0, the equation has one repeated real root (a double root). The parabola touches the x-axis at exactly one point.

若 Δ = 0，方程有一个重根（二重根）。抛物线与 x 轴相切于一点。
If Δ < 0, the equation has no real roots. The parabola never crosses the x-axis.

若 Δ < 0，方程无实数根。抛物线不与 x 轴相交。

For example, in x² – 4x + 4 = 0, the discriminant is 16 – 16 = 0, confirming a double root at x = 2. In x² + 2x + 5 = 0, the discriminant is 4 – 20 = -16, so no real roots exist.

例如，在 x² – 4x + 4 = 0 中，判别式为 16 – 16 = 0，证实 x = 2 为重根。在 x² + 2x + 5 = 0 中，判别式为 4 – 20 = -16，因此无实数根。

7. Graphical Interpretation | 图像法解读

The graph of y = ax² + bx + c is a parabola. Solving ax² + bx + c = 0 is equivalent to finding the x-coordinates where this parabola intersects the x-axis.

y = ax² + bx + c 的图像是抛物线。解方程 ax² + bx + c = 0 等价于求该抛物线与 x 轴交点的横坐标。

If a > 0, the parabola opens upward (a U shape); if a < 0, it opens downward (an inverted U shape). The vertex is the turning point, located at x = -b/(2a).

若 a > 0，抛物线开口向上（U 形）；若 a < 0，开口向下（倒 U 形）。顶点是转向点，位于 x = -b/(2a) 处。

For example, y = x² – 2x – 3 crosses the x-axis where x = -1 and x = 3, matching the solutions of x² – 2x – 3 = 0. The vertex lies midway between the roots at x = 1.

例如，y = x² – 2x – 3 与 x 轴交于 x = -1 和 x = 3，这与方程 x² – 2x – 3 = 0 的解一致。顶点位于两根正中间 x = 1 处。

If the equation has no real roots, the graph never touches the x-axis; the discriminant being negative confirms this visually. Drawing a quick sketch can help you verify your algebraic solutions.

若方程无实数根，图像不与 x 轴相交；判别式为负也证实了这一点。快速画图有助于验证代数解。

8. Solving Word Problems | 应用题求解

Many examination questions present quadratics in context, such as area, distance, speed, or number problems. The key is to define a variable, form an equation, and solve it for the relevant value.

许多考试题以实际背景呈现二次方程，如面积、距离、速度或数字问题。关键在于设定变量、建立方程并求解相关值。

Example: A rectangle has length 3 cm longer than its width. Its area is 40 cm². Let the width be x; then the length is x + 3, and the area equation is x(x + 3) = 40.

示例：一个矩形长比宽多 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3，面积方程为 x(x + 3) = 40。

Expanding and rearranging gives x² + 3x – 40 = 0, which factorises to (x + 8)(x – 5) = 0. The solutions are x = -8 or x = 5. Since a width cannot be negative, the width is 5 cm and the length is 8 cm.

展开并整理得 x² + 3x – 40 = 0，因式分解为 (x + 8)(x – 5) = 0。解为 x = -8 或 x = 5。由于宽度不能为负，因此宽为 5 cm，长为 8 cm。

Always check whether your answers make sense in the original context. Negative lengths, times, or quantities are usually discarded even though they satisfy the algebra.

务必检查答案在原始情境中是否合理。负数长度、时间或数量即使满足代数关系，通常也应舍弃。

9. Common Mistakes | 常见错误

Understanding typical errors helps you avoid them in the examination. The most frequent mistakes are related to signs, missing solutions, and failing to rearrange into standard form.

了解典型错误有助于在考试中避免它们。最常见的错误涉及符号、漏解以及未能整理成标准形式。

Forgetting ± when taking square roots: x² = 9 gives x = ±3, not just x = 3. Always include both roots.

开平方时忘记 ±：x² = 9 应得 x = ±3，而非仅 x = 3。务必包含两个根。
Sign errors when substituting into the formula: If b = -4, then -b = 4 and b² = 16. Carefully track negative signs.

代入公式时符号错误：若 b = -4，则 -b = 4，b² = 16。请仔细追踪负号。
Dividing by a variable that could be zero: Never divide both sides by x unless x ≠ 0 is guaranteed; you may lose a root.

除以可能为零的变量：除非确保 x ≠ 0，否则切勿两边同除以 x，否则可能丢根。
Forgetting to set the equation to zero before factorising: x(x + 3) = 40 is not factorised correctly as (x + something)(x + something) = 0; rearrange first.

因式分解前忘记将方程化为零：x(x + 3) = 40 不能直接当作 (x + 某)(x + 某) = 0 来分解；应先将方程化为零。

10. Examination Tips | 考试技巧

In the IGCSE examination, choose the quickest reliable method for each question. Factorisation is best for simple equations; the formula or completing the square is safer when the roots are irrational or the equation is complex.

在 IGCSE 考试中，为每道题选择最快且可靠的方法。简单方程适合因式分解；当根为无理数或方程复杂时，公式法或配方法更稳妥。

Always write down the coefficients a, b, c before substituting into the formula. This reduces careless errors.

代入公式前，先写出系数 a、b、c。这能减少粗心错误。
If the question says “give your answer correct to 2 decimal places”, use the quadratic formula and round at the end.

若题目要求”答案精确到 2 位小数”，使用二次公式并在最后四舍五入。
Check solutions by substituting back into the original equation. This takes only a few seconds and catches most errors.

通过回代原方程检验解。这只需几秒钟，却能捕捉到大多数错误。
When a quadratic has no real roots, state “no real roots” rather than trying to force an answer.

当二次方程无实数根时，直接写明”无实数根”，不要强行凑答案。

11. Practice Questions | 练习题目

Attempt these problems on your own before checking the answers below. They cover all three methods and a word problem.

请先独立尝试以下题目，再核对答案解析。题目涵盖三种解法及一道应用题。

Question 1: Solve x² – 7x + 12 = 0 by factorisation.

题目 1：用因式分解法求解 x² – 7x + 12 = 0。
Question 2: Solve 2x² + 5x – 3 = 0 using the quadratic formula, giving answers correct to 3 significant figures.

题目 2：用二次公式求解 2x² + 5x – 3 = 0，答案精确到 3 位有效数字。
Question 3: Solve x² – 6x + 2 = 0 by completing the square, leaving your answers in surd form.

题目 3：用配方法求解 x² – 6x + 2 = 0，答案保留根式形式。
Question 4: The product of two consecutive positive integers is 156. Find the integers.

题目 4：两个连续正整数的乘积为 156，求这两个整数。

Answers: (1) x = 3 or x = 4. (2) x = 0.5 or x = -3. (3) x = 3 ± √7. (4) 12 and 13.

答案：（1）x = 3 或 x = 4。（2）x = 0.5 或 x = -3。（3）x = 3 ± √7。（4）12 和 13。

12. Summary | 总结

Quadratic equations are solved by factorisation, completing the square, or the quadratic formula. The discriminant Δ = b² – 4ac reveals how many real roots exist, and the graph of the parabola visually confirms these results.

二次方程可通过因式分解、配方或二次公式求解。判别式 Δ = b² – 4ac 揭示实数根的个数，抛物线图像则可直观验证这些结果。

Method	Best Used When	Example Result
Factorisation	Simple integer roots	x = 2, x = 5
Completing the Square	Surd form or vertex required	x = 1 ± √3
Quadratic Formula	Any quadratic, especially non-factorisable	x = (−b ± √Δ) / (2a)

Master these three methods, interpret the discriminant correctly, and always verify your answers. With regular practice, solving quadratic equations becomes a routine skill that earns reliable marks.

掌握这三种方法、正确解读判别式并始终验证答案。通过规律练习，解二次方程将成为轻松拿分的常规技能。

Quick Revision Checklist: standard form → choose method → solve accurately → check roots → state final answer clearly.

快速复习清单：标准形式 → 选择方法 → 精确求解 → 检验根 → 清晰写出最终答案。

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Solving Quadratic Equations: Factorisation, Formula, and Graphs | 解二次方程：因式分解、公式法与图像法