Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear in nearly every IGCSE Mathematics examination. Mastering the methods for solving them is essential for achieving top grades. In this revision guide, we will explore factorisation, the quadratic formula, completing the square, the discriminant, and practical problem-solving, with clear bilingual explanations at every step.

一元二次方程几乎出现在每一份 IGCSE 数学试卷中。掌握其解法是取得高分的关键。在本复习指南中，我们将系统讲解因式分解法、求根公式、配方法、判别式以及实际应用题，每一步都配有清晰的中英双语解释。

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 x is 2, which gives the equation its name.

一元二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数是 2，因此得名”二次”。

The coefficient a must not be zero; otherwise the equation becomes linear. | 系数 a 不能为零，否则方程就变成了一次方程。
Examples: x² – 5x + 6 = 0, 2x² + 3x – 1 = 0, x² = 9. | 例如：x² – 5x + 6 = 0，2x² + 3x – 1 = 0，x² = 9。

2. Factorisation Method | 因式分解法

Factorisation is often the quickest method when the quadratic expression factorises nicely. The idea is to rewrite ax² + bx + c as a product of two linear brackets, then set each bracket equal to zero.

当二次表达式可以顺利分解时，因式分解法通常是最快捷的方法。其核心思想是将 ax² + bx + c 改写为两个一次括号的乘积，然后令每个括号等于零。

For example, solve x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5. These are -2 and -3, so we write (x – 2)(x – 3) = 0. Hence x = 2 or x = 3.

例如，解 x² – 5x + 6 = 0。我们寻找两个数相乘为 6，相加为 -5。这两个数是 -2 和 -3，因此可以写成 (x – 2)(x – 3) = 0。所以 x = 2 或 x = 3。

Always expand your brackets to check the factorisation is correct. | 务必展开括号检验因式分解是否正确。
If the coefficient of x² is not 1, use the method of splitting the middle term or trial and error. | 若 x² 的系数不是 1，可使用拆中项法或试错法。

3. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula provides a universal solution. For ax² + bx + c = 0, the solutions are given by the formula below.

当因式分解困难或无法进行时，求根公式提供了一种通用的解法。对于 ax² + bx + c = 0，其解由下面的公式给出。

x = (-b ± √(b² – 4ac)) / (2a)

To use the formula, substitute the values of a, b and c carefully, then simplify the result. Remember to calculate both the positive and negative cases of the ± sign.

使用公式时，要仔细代入 a、b、c 的值，然后化简结果。记得分别计算 ± 号的正负两种情况。

Write down a, b, c first to avoid confusion. | 先写出 a、b、c 的值，避免混淆。
Use a calculator only for the final arithmetic, not for the algebraic setup. | 只使用计算器进行最后的数值运算，而不是用于代数式的建立。

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 without the quadratic formula.

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不使用求根公式解方程时特别有用。

For x² + bx + c, add and subtract (b/2)² to create a perfect square. For example, x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7.

对于 x² + bx + c，通过加上再减去 (b/2)² 来构造完全平方。例如，x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7。

The turning point of the graph y = (x + p)² + q is (-p, q). | 图像 y = (x + p)² + q 的顶点坐标为 (-p, q)。
Solving (x + p)² = k gives x = -p ± √k, provided k ≥ 0. | 解 (x + p)² = k 得 x = -p ± √k，前提是 k ≥ 0。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² – 4ac. It tells us how many real roots a quadratic equation has without solving it fully.

判别式是求根公式中根号下的部分：Δ = b² – 4ac。它能在不解方程的情况下告诉我们一元二次方程有几个实数根。

Discriminant \| 判别式	Number of real roots \| 实数根个数
Δ > 0	Two distinct real roots \| 两个不同的实数根
Δ = 0	One repeated real root \| 一个重实数根
Δ < 0	No real roots \| 没有实数根

If Δ is a perfect square, the quadratic factorises over integers. If not, the roots are irrational or complex.

若 Δ 是完全平方数，则二次式可以整数因式分解；否则根为无理数或复数。

6. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots can be found directly from the coefficients.

对于一元二次方程 ax² + bx + c = 0，若其两根为 α 和 β，则根的和与积可以直接由系数得出。

Sum of roots: α + β = -b/a
Product of roots: αβ = c/a

This relationship is useful when constructing a quadratic equation from known roots, or when checking your solutions.

这个关系在由已知根构造一元二次方程，或检验解是否正确时非常有用。

If roots are m and n, the equation can be written as x² – (m + n)x + mn = 0. | 若根为 m 和 n，方程可写成 x² – (m + n)x + mn = 0。

7. Solving by Drawing Graphs | 图像法解方程

A quadratic equation can be solved graphically by finding the x-coordinates where the curve y = ax² + bx + c crosses the x-axis. These intersection points are the real roots.

可以通过绘制曲线 y = ax² + bx + c 与 x 轴的交点，来图解一元二次方程。交点的 x 坐标即为实数根。

If the curve touches the x-axis at one point, there is one repeated root. | 若曲线与 x 轴仅相切于一点，则有一个重根。
If the curve does not touch the x-axis, there are no real roots. | 若曲线不与 x 轴相交，则无实数根。
Graphical solutions are often approximate; use algebraic methods for exact answers. | 图像解通常是近似值，要求精确答案时应使用代数方法。

8. Word Problems | 应用题

Many real-world problems lead to quadratic equations. Common scenarios include area problems, projectile motion, and consecutive number puzzles. The strategy is to translate the words into a quadratic equation, solve it, and then check whether each root makes sense in context.

许多实际问题都会引出一元二次方程。常见情境包括面积问题、抛体运动以及连续整数谜题。解题策略是将文字转化为二次方程，求解，然后检查每个根在具体情境中是否有意义。

For example: The area of a rectangle is 30 cm² and its length is 3 cm longer than its width. Let the width be x, then x(x + 3) = 30, so x² + 3x – 30 = 0. The positive root gives the width, while the negative root is rejected because a length cannot be negative.

例如：一个矩形的面积为 30 平方厘米，长比宽多 3 厘米。设宽为 x，则 x(x + 3) = 30，即 x² + 3x – 30 = 0。正根给出宽度，负根因长度不能为负而被舍去。

9. Quadratic Inequalities | 二次不等式

Solving a quadratic inequality such as x² – 5x + 6 < 0 involves finding the critical roots first, then testing intervals on a number line. The solution is often a range of x values.

解二次不等式（如 x² – 5x + 6 < 0）时，首先要找到临界根，然后在数轴上检验区间。解通常是一个 x 的取值范围。

For x² – 5x + 6 < 0, the roots are x = 2 and x = 3. Because the parabola opens upward, the inequality holds between the roots, so the solution is 2 < x < 3.

对于 x² – 5x + 6 < 0，根为 x = 2 和 x = 3。由于抛物线开口向上，不等式在两根之间成立，因此解为 2 < x < 3。

Remember to reverse the inequality sign if you multiply or divide by a negative number. | 记得如果乘以或除以负数，不等号方向要反转。

10. Common Mistakes | 常见错误

Students often lose marks on quadratic equations due to a few repeated errors. Being aware of these can help you avoid them in the exam.

学生在解一元二次方程时常因几个反复出现的错误而失分。了解这些错误有助于你在考试中避免它们。

Forgetting to rearrange the equation to the form ax² + bx + c = 0 before factorising or using the formula. | 在因式分解或使用公式前，忘记将方程整理为 ax² + bx + c = 0 的形式。
Misstating the signs of b and c when substituting into the quadratic formula. | 代入求根公式时弄错 b 和 c 的符号。
Dropping one solution when solving an equation like x² = 9; remember x = ±3. | 解 x² = 9 时漏掉一个解；记住 x = ±3。
Dividing both sides of an equation by x, which loses the root x = 0. | 方程两边同时除以 x，从而丢失了 x = 0 这个根。

11. Practice Questions | 练习题

Test your understanding with these questions. Try to solve them without a calculator first, then check your answers.

用以下问题测试你的理解。先尝试不使用计算器求解，然后核对答案。

Solve x² – 7x + 12 = 0 by factorisation. | 用因式分解法解 x² – 7x + 12 = 0。
Solve 2x² + 5x – 3 = 0 using the quadratic formula. | 用求根公式解 2x² + 5x – 3 = 0。
Find the value of the discriminant for x² + 4x + 5 = 0, and state the number of real roots. | 求 x² + 4x + 5 = 0 的判别式的值，并说明实数根的个数。
Complete the square for x² + 8x + 1, and hence solve the equation. | 对 x² + 8x + 1 配方，并由此解方程。

Answers: 1) x = 3 or 4. 2) x = 0.5 or -3. 3) Δ = -4, no real roots. 4) (x + 4)² – 15 = 0, so x = -4 ± √15.

答案：1) x = 3 或 4。2) x = 0.5 或 -3。3) Δ = -4，无实数根。4) (x + 4)² – 15 = 0，因此 x = -4 ± √15。

12. Conclusion | 总结

Quadratic equations are a fundamental part of IGCSE Mathematics. By mastering factorisation, the quadratic formula, completing the square, and the discriminant, you can approach any quadratic problem with confidence. Regular practice is the key to success.

一元二次方程是 IGCSE 数学的重要组成部分。通过掌握因式分解、求根公式、配方法和判别式，你可以自信地应对任何相关的二次方程问题。坚持练习是成功的关键。

Remember to always check whether your solutions are reasonable, especially in word problems. Write out each step clearly to avoid sign errors, and use the discriminant to predict the nature of the roots before solving.

记住，要始终检查你的解是否合理，尤其是在应用题中。清晰写出每一步以避免符号错误，并在求解前用判别式预判根的性质。

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Discriminant \| 判别式	Number of real roots \| 实数根个数
Δ > 0	Two distinct real roots \| 两个不同的实数根
Δ = 0	One repeated real root \| 一个重实数根
Δ < 0	No real roots \| 没有实数根

Solving Quadratic Equations | 解一元二次方程